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A generalized acceptance urn model

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A generalized acceptance urn model
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Wagner, Kevin
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Lattice paths
Rotation method
Ballot problem
Bayesian approach
Ruin problem
Generalized binomial series
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
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Abstract:
ABSTRACT: An urn contains two types of balls: p "+t" balls and m "-s" balls, where t and s are positive real numbers. The balls are drawn from the urn uniformly at random without replacement until the urn is empty. Before each ball is drawn, the player decides whether to accept the ball or not. If the player opts to accept the ball, then the payoff is the weight of the ball drawn, gaining t dollars if a "+t" ball is drawn, or losing s dollars if a "-s" ball is drawn. We wish to maximize the expected gain for the player. We find that the optimal acceptance policies are similar to that of the original acceptance urn of Chen et al. with s=t=1. We show that the expected gain function also shares similar properties to those shown in that work, and note the important properties that have geometric interpretations. We then calculate the expected gain for the urns with t/s rational, using various methods, including rotation and reflection. For the case when t/s is irrational, we use rational approximation to calculate the expected gain. We then give the asymptotic value of the expected gain under various conditions. The problem of minimal gain is then considered, which is a version of the ballot problem. We then consider a Bayesian approach for the general urn, for which the number of balls n is known while the number of "+t" balls, p, is unknown. We find formulas for the expected gain for the random acceptance urn when the urns with n balls are distributed uniformly, and find the asymptotic value of the expected gain for any s and t. Finally, we discuss the probability of ruin when an optimal strategy is used for the (m,p;s,t) urn, solving the problem with s=t=1. We also show that in general, when the initial capital is large, ruin is unlikely. We then examine the same problem with the random version of the urn, solving the problem with s=t=1 and an initial prior distribution of the urns containing n balls that is uniform.
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Dissertation (Ph.D.)--University of South Florida, 2010.
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ABSTRACT: An urn contains two types of balls: p "+t" balls and m "-s" balls, where t and s are positive real numbers. The balls are drawn from the urn uniformly at random without replacement until the urn is empty. Before each ball is drawn, the player decides whether to accept the ball or not. If the player opts to accept the ball, then the payoff is the weight of the ball drawn, gaining t dollars if a "+t" ball is drawn, or losing s dollars if a "-s" ball is drawn. We wish to maximize the expected gain for the player. We find that the optimal acceptance policies are similar to that of the original acceptance urn of Chen et al. with s=t=1. We show that the expected gain function also shares similar properties to those shown in that work, and note the important properties that have geometric interpretations. We then calculate the expected gain for the urns with t/s rational, using various methods, including rotation and reflection. For the case when t/s is irrational, we use rational approximation to calculate the expected gain. We then give the asymptotic value of the expected gain under various conditions. The problem of minimal gain is then considered, which is a version of the ballot problem. We then consider a Bayesian approach for the general urn, for which the number of balls n is known while the number of "+t" balls, p, is unknown. We find formulas for the expected gain for the random acceptance urn when the urns with n balls are distributed uniformly, and find the asymptotic value of the expected gain for any s and t. Finally, we discuss the probability of ruin when an optimal strategy is used for the (m,p;s,t) urn, solving the problem with s=t=1. We also show that in general, when the initial capital is large, ruin is unlikely. We then examine the same problem with the random version of the urn, solving the problem with s=t=1 and an initial prior distribution of the urns containing n balls that is uniform.
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Advisor: Stephen Suen, Ph.D.
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Lattice paths
Rotation method
Ballot problem
Bayesian approach
Ruin problem
Generalized binomial series
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Doctoral.
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AGeneralizedAcceptanceUrnModel by KevinP.Wagner Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematicsandStatistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:StephenSuen,Ph.D. GregoryMcColm,Ph.D. NatasaJonoska,Ph.D. BrendanNagle,Ph.D. DateofApproval: April5,2010 Keywords:latticepaths,rotationmethod,ballotproblem,Bayesianapproach,ruinproblem, generalizedbinomialseries c Copyright2010,KevinP.Wagner

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TableofContents ListofFigures iii ListofTables v Abstract vi Chapter1.TheAcceptanceUrnModel1 1.1.IntroductionandMainResults1 1.2.ApplicationoftheUrnModel6 1.3.AHistoryoftheUrnModel6 1.4.TheGeneralizedAcceptanceModel9 Chapter2.Propertiesof G m;p ; s;t 13 2.1.AHierarchyofInequalities16 2.2.Continuityof G m;p ; s;t in s and t 21 2.3.MiscellaneousResults22 Chapter3.The m;p ;1 ;t Urns27 3.1.GeneralizedBinomialSeries27 3.2.The m;p ;1 ; 1Urns30 3.3.TheValueofthe m;p ;1 ;t Urn33 3.3.1.The m;p ;1 ;t Urnswith m pt 36 3.3.2.OtherStrategies42 3.3.3.The m;p ;1 ;t Urnswith m 192 5.6.Urnswith pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms 0and p p = o pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms 93 i

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Chapter6.TheProbabilityofMinimalGain94 6.1.ABriefHistoryoftheBallotProblem94 6.2.ZeroGainforthe m;p ;1 ;t Urns, t 0anInteger95 6.3.GeneralizedBallotNumbersandZeroGainNumbers96 6.4.AnotherExtensionoftheBallotProblem103 6.5.TheFirstCrossing107 Chapter7.ABayesianApproach109 7.1.Preliminaries109 7.2.DeterminingtheBettingRule111 7.3.TheOriginalRandomUrn112 7.4.AnAlgorithmfortheValue G n; ;1 ; 1118 7.5.AFair n; ; s;t Urn120 7.6.The n; ; s;t Urnswith Uniform121 7.7.AnAlgorithmfortheGeneralRandomUrn126 Chapter8.ARuinProblem130 8.1.TheRuinProblemforthe m;p ; s;t Urn130 8.1.1.TrendsandPreliminariesforthe m;p ;1 ; 1and m;p ;1 ;t Urns131 8.1.2.GeneralResults134 8.2.SolutionoftheRuinProblemforthe m;p ;1 ; 1Urns136 8.2.1.RuinwiththeZero-BetStrategy137 8.2.2.RuinwiththeZero-PassStrategy139 8.3.TheRuinProblemfortheRandomAcceptanceUrn140 8.3.1.TheRuinProbabilitywith s = t =1140 8.3.2.ASaferOptimalStrategy141 Chapter9.Conclusion144 ListofReferences 149 AbouttheAuthorEndPage ii

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ListofFigures Figure1.1.Agraphicalrepresentationoftherealization = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2,+3, )]TJ/F8 9.9626 Tf 7.749 0 Td [(2, )]TJ/F8 9.9626 Tf 7.749 0 Td [(2,+3, )]TJ/F8 9.9626 Tf 7.749 0 Td [(2,+3,fromthe ; 3;2 ; 3urn.10 Figure2.1.Arealization fromthe ; 3;2 ; 3urnblack,andthecorresponding realizationfromthe ; 4;3 ; 2antiurnpink.14 Figure2.2.Fortherealization fromthe ; 3;1 ; 1urnsolid,rotationaboutthe point P ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1producesthereversedrealization R dashed.16 Figure2.3.Identicationofballsforthe m;p +1; s;t urntop,themodied m;p ; s;t urnwithablankballcenter,andthe m +1 ;p ; s;t urn bottom.17 Figure3.1.Forthisrealizationfromthe ; 8;1 ; 1urnblack,thedashedportion isa )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripfromneutral,andresultsinagainofonewhenthe zero-betstrategyisused.31 Figure3.2.Segmentidenticationofarealizationfromthe ; 4;1 ; 2urnwith N 2.37 Figure3.3.Segmentidenticationofarealizationfromthe ; 4;1 ; 2urnwith N + 2.40 Figure3.4.Arealizationwithcrossingnumber i> 0,andtherealizationwith crossingnumber i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1associatedwithit.44 Figure4.1.Forthisrealizationfromthe ; 4;1 ; 2urn,thethird,sixth,andtwelfth ballsblackresultinup-crossingswhenthezero-passstrategyisused, witheachresultinganagainofonefortheplayer.51 Figure4.2.Forthisrealization fromthe ; 5;2 ; 3urn,thedashedportionsform + ,andthesolidportionsform )]TJ/F8 9.9626 Tf 10.046 -3.615 Td [(whenthezero-betstrategyisused.63 Figure6.1.Azero-passzero-gainrealizationfromthe ; 4;2 ; 3urnandits corresponding ; 7;3 ; 2ballotpermutationreectedthroughthe n -axisandshiftedleftoneunit.101 Figure6.2.For s =8, t =3,wehave r =0, f = ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0,andthe piecewiseboundary @ f .106 Figure7.1. a i;j and P j for n =4and uniform.119 Figure7.2. c i;j for n =4and uniform.120 Figure7.3. P j and a i;j fortheinitialdistribution with p 0 = p 4 =1 = 20, p 1 = p 3 =1 = 4,and p 2 =2 = 5.120 iii

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Figure7.4.Bettingtreeandbettinglinefora ; ;5 ; 3urn,with uniform.122 Figure7.5. a i;j and P j for n =3and uniform.126 Figure7.6. c i;j ,^ c i;j smallertableatright,and q i; 0, i =1 ; 2 ; 3for s =2, t =3, n =3,and uniform.127 Figure7.7.Thealgorithmappliedtothecase s =9, t =4, n =3and uniform.128 Figure8.1.Mappingofazero-passruinrealizationblacktoazero-betruin realizationshadowwith b =1onthe ; 7;1 ; 1urn.133 iv

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ListofTables Table3.1.Distributionofthegainforthe m;p ;1 ; 2urns,with m + p =5.48 Table4.1.Determinationofthecrossingconstants a i b i ,and c i for )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 i 4, with s =4and t =3.54 Table4.2.Expectedgainforsome ; ;4 ; 3urns,roundedtotwodecimalplaces.57 Table6.1. ; 3-ballotnumbersfor0 a 11,0 b 7.97 Table6.2. ; 3-zero-gainnumbersfor0 a 11,0 b 7.98 Table8.1.Some ; ;2 ; 3urnruinprobabilitiesinpercentwithvetotalballsand thezero-betstrategy.136 Table8.2.Some ; ;2 ; 3urnruinprobabilitiesinpercentwithvetotalballsand thezero-passstrategy.137 Table8.3.Probabilitiesofruinapproximatedtothreedecimalplaces,with uniform,usingthezero-betstrategy.141 Table8.4.Probabilitiesofruinapproximatedtothreedecimalplaces,with uniform,usingthezero-passstrategy.143 v

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AGeneralizedAcceptanceUrnModel KevinP.Wagner ABSTRACT Anurncontainstwotypesofballs: p + t "ballsand m )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls,where t and s arepositive realnumbers.Theballsaredrawnfromtheurnuniformlyatrandomwithoutreplacementuntilthe urnisempty.Beforeeachballisdrawn,theplayerdecideswhethertoaccepttheballornot.Ifthe playeroptstoaccepttheball,thenthepayoistheweightoftheballdrawn,gaining t dollarsifa + t "ballisdrawn,orlosing s dollarsifa )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "ballisdrawn.Wewishtomaximizetheexpected gainfortheplayer. Wendthattheoptimalacceptancepoliciesaresimilartothatoftheoriginalacceptanceurn ofChen etal. [7]with s = t =1.Weshowthattheexpectedgainfunctionalsosharessimilar propertiestothoseshowninthatwork,andnotetheimportantpropertiesthathavegeometric interpretations.Wethencalculatetheexpectedgainfortheurnswith t=s rational,usingvarious methods,includingrotationandreection.Forthecasewhen t=s isirrational,weuserational approximationtocalculatetheexpectedgain.Wethengivetheasymptoticvalueoftheexpected gainundervariousconditions.Theproblemofminimalgainisthenconsidered,whichisaversion oftheballotproblem. WethenconsideraBayesianapproachforthegeneralurn,forwhichthenumberofballs n is knownwhilethenumberof+ t "balls, p ,isunknown.Wendformulasfortheexpectedgain fortherandomacceptanceurnwhentheurnswith n ballsaredistributeduniformly,andndthe asymptoticvalueoftheexpectedgainforany s and t Finally,wediscusstheprobabilityofruinwhenanoptimalstrategyisusedforthe m;p ; s;t urn, solvingtheproblemwith s = t =1.Wealsoshowthatingeneral,whentheinitialcapitalislarge, ruinisunlikely.Wethenexaminethesameproblemwiththerandomversionoftheurn,solving theproblemwith s = t =1andaninitialpriordistributionoftheurnscontaining n ballsthatis uniform. vi

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1. TheAcceptanceUrnModel 1.1. IntroductionandMainResults. The acceptanceurnproblem isasfollows:Anurncontains p ballsofvalue+ t ,and m ballsofvalue )]TJ/F11 9.9626 Tf 7.748 0 Td [(s .Theballsaredrawnoutfromtheurnuniformlyat random,withoutreplacement,untiltheurnisempty.Beforeeachdrawfromtheurn,aplayeris askedtodecidewhetherhewouldliketoacceptthenextball,withthepayoutbeingthevalueof thenextballdrawnfromtheurn.Theproblemistondanoptimalstrategythatmaximizesthe player'sexpectedgain,andtocalculatetheexpectedgain.Theoriginalacceptanceurnmodelwith s = t =1wasproposedinChen etal. [7].Inthiswork,wegeneralizetheproblemsothat s and t maybearbitrarynonnegativerealnumbers,ndinganoptimalstrategyandcalculating theexpectedgain.Wendthatanoptimalstrategydependsonlyonthesignofthe weight ofthe urn,thatis,thecombinedvalueoftheballsleftintheurnafter j ballshavebeendrawn,where 0 j
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directmanipulationoftheexpectedgainfunction.1.Wealsondthattheexpectedgainwill notdecreaseif t isincreased, s isdecreased,ortheballvaluesareincreasedattheexpenseofthe numberofballs.Inparticular,weshowthattheexpectedgainfunctioniscontinuousinboth s and t ,whichenablesustouserationalapproximationtocalculatetheexpectedgainwhen t=s is irrational. WeworktowardndingformulasexpressingtheexpectedgaininChapters3and4.Forthe originalmodelwith s = t =1,Chen[7,Theorem2.2]showedthattheexpectedgainwith m )]TJ/F8 9.9626 Tf 7.748 0 Td [(1" ballsand p +1"ballsusinganoptimalstrategyequals .1max f 0 ;p )]TJ/F11 9.9626 Tf 9.962 0 Td [(m g + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 min f m;p g)]TJ/F7 6.9738 Tf 10.309 0 Td [(1 X k =0 m + p k : Weprovideamoredirectproofofthisresult,byshowingthatformax f 0 ;p )]TJ/F11 9.9626 Tf 10.586 0 Td [(m g k p ,the probabilitytheplayergainsatleast k equals m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Weuseareectionmethodtoshowthisresult. Generalizing,wendthatreectionisnotausefulmethodforcalculatingtheexpectedgainin general.However,wendthatthereissignicantrotationalsymmetrywhen t isanintegermultiple of s ,orviceversa.ApplyingtherotationmethodindicatedbytheReversalLemma,weshowa fundamentalresultwiththe CrossingLemma Lemma3.7.Aversionofthisresultappearedin Mohanty[15,eq.]usingtheconvolutionofpaths,butwithouttheuseofrotation. Mohanty'smotivationwastocountthenumberofpathswithaparticularproperty,whileinthis workourinterestisintheactualbijectionthatwasindicatedin[15],andusingitasatooltocount pathswithotherproperties.UsingtheresultoftheCrossingLemma,weareabletogivemultiple expressionsfortheexpectedgain.When m pt ,theseincludethe zerocount form, m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t 2 p X k =1 kt + k k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ; whichiscalculatedbycountingthenumberoftimestheurnis neutral ,thatis,thenumberoftimes theurnweightiszero,bylocation.Thiswasthemethodoriginallyusedin[7]alongwiththeuse ofacombinatorialidentitytoshow.1.Adierentmethodofcounting,againwiththeaidofthe 2

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CrossingLemma,producesthe negativebinomial formfor m pt m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 m + k k t +1 p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k ; andamodicationofthismethodproducesthe binomialform ,againfor m pt : )]TJ/F11 9.9626 Tf 9.636 6.74 Td [(t 2 + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p X k =0 m + p +1 k t p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Wealsoindicatehowtocalculatetheprobabilitythattheplayerwillgainatleast ` usinganoptimal strategy,forany m p ,and ` ,usingarotationargument.Thisresultsinanotherformulaforthe expectedgain,the distributionform ,whichreducesto.1when s = t =1. Ingeneral,themethodsusedtoderivetheformulasabovedonotworkwhen s and t arearbitrary, astheresultoftheCrossingLemmadoesnotholdforageneral s and t .Usingamoreprimitive approachinChapter4,weareabletogiveanotherformula,the crossingsform ,fortheexpectedgain when s =1and t isapositiveinteger.Thismethodofcalculationcanbegeneralizedsothat s may alsobeaninteger,andtheresultisarathercomplicatedformulawhencomparedtotheformulas abovefortheexpectedgain,forthecasewhen t=s isrational.Fortheurnswith ms = pt ,weshow thatthisformulacanbesimplied.When t=s isirrational,thecontinuityoftheexpectedgainin s and t allowsustousearationalapproximationof t=s toobtainaformulafortheexpectedgain. Finally,weshowthatndingthedistributionofthegainisequivalenttondingthedistributionof themaximumurnweightachievedduringplay.Thisresult,liketheearliercasewith s =1and t a positiveinteger,isshownwiththehelpofarotationmethod. Forlarge m and p ,weshowinChapter5thatif ms )]TJ/F11 9.9626 Tf 10.038 0 Td [(pt = p p tendstozero,thentheexpected gainisasymptoticallyequalto p 2 pt t + s 4 : Wealsondtheasymptoticvalueoftheexpectedgaininthecase ms )]TJ/F11 9.9626 Tf 9.915 0 Td [(pt = p p ,generalizing theresultofChen etal. [7,Theorem3.2].Inthecase ms )]TJ/F11 9.9626 Tf 9.544 0 Td [(pt = p p !1 and ms )]TJ/F11 9.9626 Tf 9.544 0 Td [(pt =p 0,we showthattheexpectedgainisasymptoticallyequalto t t + s 2 p ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt : When s = t =1,itwasshownin[7,Theorem3.1]thatif m=p > 1,thenasymptoticallythe expectedgainequals 1 )]TJ/F15 7.9701 Tf 6.586 0 Td [(1 = p m )]TJ/F17 7.9701 Tf 6.587 0 Td [(p .Weextendthisresulttothecasewith s =1and t apositive 3

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integer,showingthatif m= pt > 1,thentheasymptoticvalueoftheexpectedgainequals t +1 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = t t +1 2 p m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt ; usingthebinomialformoftheexpectedgain.Wethenusethisresulttoshowthatthevalueis boundedinthegeneralcasewhen ms= pt > 1,forarbitrarynonnegative s and t .Finally, when pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms = p p !1 ,weshowthattheexpectedgainisasymptoticallyequalto pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms Weexaminearelatedproblemtotheacceptanceurnproblem,the ballotproblem ,inChapter6. Fortheballotproblem,candidate A receives a votesinanelection,whilecandidate B receives b votes,with a b ,with 0axedrealnumber.Thequestionistondtheprobabilitythat,as thevotesarecounted,candidate A alwayshasmorethan timesthenumberofvotesfor B .The probabilitythattheplayermakestheminimalgainpossibleusinganoptimalstrategyisavariation onthisballotproblem.Thereectionmethodthatweuseinthisworkdatesbacktothesolutionof theballotproblemwith =1.See,forexample,Feller[9]andRenault[19].Forthe casewith apositiveinteger,thesolutiontotheproblemusingarotationmethodwasshownin GouldenandSerrano[11].Forarbitrary ,thesolutiontotheproblemwasgivenbyTakacs [24],thoughingeneralthesolutionisnotexplicit.Wediscuss generalizedballotnumbers and generalizedzerogainnumbers ,andinparticular,ndexplicitsolutionsforboththeballotandzero gainproblemsforthecasewhen isthereciprocalofaninteger.Thegeneralizationoftheballot problemofIrvingandRattan[12],whichusesarotationmethod,anditsapplicationtothe acceptanceurnmodelisalsodiscussed. InChapter7,weconsidertheBayesianversionoftheurn,wherethetotalnumberofballs n is known,butthenumberof+ t "balls, p ,israndomandisdeterminedbyaninitialpriordistribution on f 0 ;:::;n g .Wedevelopaconditionthat,whenmet,indicateswhentheplayershallaccept thenextballdrawnfromtheurn.In[7],itwasdeterminedthatanoptimalacceptancestrategy forthecase s = t =1and uniformisacceptifandonlyifatleastasmany`+1'ballsas` )]TJ/F8 9.9626 Tf 7.748 0 Td [(1' ballshavebeendrawn."Inthiswork,wendalargerfamilyofinitialpriordistributionswith s = t =1forwhichthisruleisoptimal.Forthisfamilyofdistributions,wegiveaformulafor theexpectedgain,andndthattheplayerattemptstogaintheinitialweightoftheurn,ifitis positive.Wealsondasecondfamilyofdistributionsforwhichtheoppositeruleisoptimal:bet ifandonlyifatleastasmany` )]TJ/F8 9.9626 Tf 7.748 0 Td [(1'ballsas`+1'ballshavebeendrawn."Withthissecondfamily ofdistributions,wendthattheplayeraimstotakeadvantageofuctuationsthatwilloccuralong 4

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theaway,insteadoftryingtocapturetheinitialweightoftheurnwhenpositive.Wegivea formulafortheexpectedgainforthesedistributionsaswell.Forotherdistributions,wedevelopan algorithmthatgivesanindicationofwhenandwhennottobet,thatalsocalculatestheexpected gain.However,thealgorithmisnotveryecient,andingeneralthealgorithmrequiresasignicant amountofinformationtobeknownbeforehand. Forarbitrary s and t ,wendthatwhen isabinomialdistributionwith n trialsandprobability ofsuccess s= t + s ,therandomacceptanceurnisafairgame.Thatis,anyacceptancestrategyis optimal,andtheexpectedgainequalszero.When isuniform,wendarelativelysimpleoptimal acceptancepolicy,calculatetheexpectedgain,andndthatwhen n islargetheexpectedgainis asymptoticallyequalto nt 2 2 t + s .Wemodifythealgorithmusedfortherandomurnswith s = t =1so thatitwillalsopresentanindicationofwhentobetandtheexpectedgain,fortheurnscontaining )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "and+ t "balls,forany n and Weintroducearuinproblemthatcanbeassociatedwiththeacceptanceurnmodel,inChapter 8.Wegivetheplayerabank b ,andifatsomepointduringplaytheplayerhaslostmorethan b )]TJ/F11 9.9626 Tf 10.305 0 Td [(s ,wesaytheplayeris ruined .Forthecase s = t =1,weshowthatprobabilitytheplayeris ruinedwhileusinganoptimalstrategythatmaximizestheexpectedgaindependsonthechoiceof thatstrategy.Wesolvetheproblemwith s = t =1withthezero-betstrategyDenition1.9and zero-passstrategyDenition1.8byndingtheprobabilitythattheplayerisruinedandthebank attainsamaximumvalueof b + k beforetheplayerisruined,foreach k 0. WealsoapplytheruinproblemtotheBayesianversionoftheurn.Forthoseurns,wesolvethe problemwith s = t =1andany b foronefamilyofinitialdistributions,asthesolutionisrelatedto thecalculationoftheexpectedgain.Inparticular,weshowthatwith uniform,theprobabilityof ruintendstozeroas b !1 Inadditiontotheballotproblem,manyoftheresultsinthisworkcanbeappliedtotheareaof latticepaths.Aparticlestartsatthepoint ; 0,andateachstagemoveseitheroneunituporone unittotheright,untilitreachesthepoint p;m .Thecombinedmovementsforma latticepath .For generalinformationonlatticepaths,seeMohanty[14]andKrattenthaler[13].The resultsfortheacceptanceurnmodelwith m ballsofvalue )]TJ/F11 9.9626 Tf 7.749 0 Td [(s and p ballsofvalue+ t applytothe behavioroflatticepathsfrom ; 0to p;m relativetolineswithslope t=s .Inparticular,inSuen andWagner[23],someoftheapplicationstolatticepathswith s =1and t apositiveintegerare shown. 5

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1.2. ApplicationoftheUrnModel. Supposethatweexpectthepriceofabondorstocktogo up p timesanddown m times,withtheratiooftheamountthestockgoesuptodownequalto t=s Beforeeachupordown,theplayercaneitherbuyifthestockisnotalreadyheld,sellifthestock isheld,ordonothingwaitorhold.Shortsellingormultiplepurchasesarenotconsidered.After m + p timeperiods,thegamenishes,andtheplayersellsanystockheld.Wewishtomaximizethe player'sexpectedgain. Thetranslationofthisproblemtotheacceptanceurnmodelisasfollows:Eachbuy"Bsignies achangefrompass"toaccept,"eachsell"Ssigniesachangefromaccept"topass,"while wait"Worhold"Hsigniesnochangefromthepreviousdecisionofpass"oraccept," respectively.Attheendofthegame,iftheplayerisholdingstock,itissold[S].Then,for example,thesequence PAAAPAPPPAAPAP, wherewherePindicatespass"andAindicatesaccept,"translatestothesequence WBHHSBSWWBHSBS[S]. ItwillbeshownwithTheorem1.6thatanoptimalstrategyistobuyorholdif s timesthenumber ofupsisatleastmorethan t timesthenumberofdowns,andsellorwaitotherwise. Ifweinsistthattheactualpriceofthestockincreasesordecreasesbythesamequantityat eachtimeperiod,thenwemayinterpretthedierencebetween s and t intermsofabuyorhold fee."Thepriceofthestockwillthengoupordown t + s 2 eachtimeperiod,buttheplayerwillgain t + s 2 + t )]TJ/F17 7.9701 Tf 6.586 0 Td [(s 2 = t orlose t + s 2 )]TJ/F17 7.9701 Tf 10.605 4.708 Td [(t )]TJ/F17 7.9701 Tf 6.587 0 Td [(s 2 = s ,ifhechoosestobuyorholdthestock.Fromtheplayer'spoint ofview,thestockwillgoup t ,ordown s ateachstage,uponinclusionofthisincentiveif t>s or feeif s
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withoutreplacementuntiltheplayerchoosestostop,whichincludestheoptionofnotdrawingat all.Theplayer'sgainwillbethecombinedweightoftheballsdrawn.W.M.Boyce[4] oeredthatthisurnmodelisaniteanalogytothefollowingbondsellingproblem:Aninvestor owns$10,000ofbondswhichmatureinthreemonths.Thecurrentmarketvalueis$10,100.Should hesellnow,selllater,orholdtomaturity? Shepp'squestionwas:whichoftheseurnsarefavorable )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(i.e.haveanexpectedgain,or value V m;p > 0 ?Sheppfoundthatthereexistsasequence ,...,suchthatthe m;p urnsfor which m p satisfy V m;p > 0,with V m;p =0forallother m .Inparticular,thenumbers p satisedthelimit lim p !1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(p p 2 p ; where 0 : 8399istheuniquerealroottotheequation .2 = )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 Z 1 0 exp )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 = 2 d: Thefavorableurnsareputintoacollection C ,andtheoptimaldrawingpolicyistodrawthe j th ballaslongastheurn,withtherst j )]TJ/F8 9.9626 Tf 10.102 0 Td [(1drawsremoved,remainsinthecollection C .Asforthe value V m;p ,dening E m;p = m m + p )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ V m )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;p + p m + p 1+ V m;p )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 wehave V m;p =max f 0 ;E m;p g : )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(See[4,Appendix1]. Itwasshownin[20,Section6]that addinga+1intotheurnneverhurts,addinga )]TJ/F8 9.9626 Tf 7.748 0 Td [(1neverhelps,andswappingouta )]TJ/F8 9.9626 Tf 7.749 0 Td [(1fora+1 neverhurts.Boyce[5,Theorem3.8]showedthataddinginbotha+1anda )]TJ/F8 9.9626 Tf 7.749 0 Td [(1neverhurts, andalsogavesometighterbounds.Deningthesecond-orderdierences 2 V p m = V m +2 ;p )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 V m +1 ;p + V m;p ; 2 V m p = V m;p +2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 V m;p +1+ V m;p ; 2 V m;p = V m +2 ;p )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 V m +1 ;p +1+ V m;p +2 ; itwasshowninChenandHwang[6]that 2 V p m 0, 2 V m p 0,and 2 V m;p 0. Dening V m;p = V m;p +1+ V m +1 ;p )]TJ/F11 9.9626 Tf 9.962 0 Td [(V m;p )]TJ/F11 9.9626 Tf 9.963 0 Td [(V m +1 ;p +1 ; 7

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itwasalsoshownthat V m;p 0,moreover,fornonnegativeintegers a and b theinequality V m + a;p + b + V m;p V m + a;p + V m;p + b holds.Asforwhat V m;p lookslikeasymptotically: V p;p )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 p p forlarge p ,where isgivenby.2.[20,eq..7] Forxed m )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [( p V m;p = p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + m p +1 + O p )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 : [5,Section6] V p;p 0for > 1andlarge p: [6,Theorem4] V p;p )]TJ/F11 9.9626 Tf 9.963 0 Td [( p for0 < 1andlarge p: [6,Theorem4] Boyce[4]proposedthe n ; P urn,aBayesianversionoftheproblem,wherethetotalnumberof balls n isxed,andtheurnswith n ballsaredistributedaccordingtosomeprobabilitydistribution P on f 0 ;:::;n g .Boycethendescribedaseriesofcomputations[4,Theorem4]thatgaveanindication ofwhentheplayershouldstop.Theoutputofthealgorithmwasthevalueoftherandomurn V n ; P Theorem1.1. Boyce Givenan n ; P urn,for 0 j n let P j = P j )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 and a n;j = b n;j =0 .For 0 j n )]TJ/F8 9.9626 Tf 9.984 0 Td [(1 ,let a n )]TJ/F8 9.9626 Tf 9.985 0 Td [(1 ;j = P j )]TJ/F11 9.9626 Tf 9.984 0 Td [(P j +1 ,andfor 0 i n )]TJ/F8 9.9626 Tf 9.985 0 Td [(2 andeach j let a i;j = a i +1 ;j + a i +1 ;j +1 : For 0 i n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 andeach j let b i;j =max 0 ;a i;j + b i +1 ;j + b i +1 ;j +1 : Thenanoptimalstoppingruleformaximizingtheexpectedscoreatthestoppingtimeis:if k balls havebeendrawn, ` ofthemminus,drawagainifandonlyif b k;` > 0 .Thevalue V n ; P ofthe n ; P urnunderoptimalplayis b ; 0 Chen etal. [7]oeredamodicationofShepp'surnmodel,the acceptanceurn .Itwasshown thatanoptimalstrategymaximizingtheexpectedgainforthisacceptanceurnistobeti.e.accept ifandonlyifthereareatleastasmany+1ballsas )]TJ/F8 9.9626 Tf 7.749 0 Td [(1ballsremainingintheurn.UnlikeShepp's urn,allbutthetrivialurnswith p =0produceavalue V m;p > 0.Infact, .3 j V m;p )]TJETq1 0 0 1 262.035 107.848 cm[]0 d 0 J 0.398 w 0 0 m 8.025 0 l SQBT/F11 9.9626 Tf 262.035 99.645 Td [(V p;m j = j m )]TJ/F11 9.9626 Tf 9.963 0 Td [(p j [7,Theorem2.1] 8

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followsbysymmetry.Theexpectedgainwasgivenexplicitlyas V m;p =max f 0 ;p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m g + m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 min f m;p g)]TJ/F7 6.9738 Tf 10.309 0 Td [(1 X i =0 m + p i : Itwasshownthat V m;p V m;p ,afactintuitivelyclearsincetheplayerhastheopportunity toresumebettingaftertherststop.Asymptotically, V m;p p m )]TJ/F17 7.9701 Tf 6.586 0 Td [(p ,if m p > 1as m !1 : [7,Theorem3.1] V m;p max f 0 ; 2 g +exp 2 = 2 R 1 j j exp )]TJ/F11 9.9626 Tf 7.748 0 Td [(t 2 = 2 dt ,ifas m !1 p )]TJ/F17 7.9701 Tf 6.586 0 Td [(m p 2 p [7,Theorem3.2] V k + p;p p p= 2as p !1 ,foranyxedinteger k: [7,Theorem3.1] Likethestoppingurn,theacceptanceurnhasasimilarstructure:addinga+1neverhurts, addinga )]TJ/F8 9.9626 Tf 7.749 0 Td [(1neverhelps,andaddingbotha+1anda )]TJ/F8 9.9626 Tf 7.749 0 Td [(1neverhurts[7,Theorems2.5and2.6].It wasalsoshownthatboth V km;m and V m;km werestrictlyincreasingin m forinteger k 1 [7,Theorem2.7].Dening 2 V m p ,etc.,aswasdonewithShepp'surn,itwasshown[7,Theorem 4.5]that 2 V p m 0, 2 V m p 0,and 2 V m;p 0.Thequadrangleinequality V m;p + V m +1 ;p +1 V m +1 ;p + V m;p +1 wasalsoshownwith[7,Theorem4.6],whiletheconcavityofthefunction f k = V m + k;p + k wasshownwith[7,Theorem4.7]: V m;p + V m +2 ;p +2 2 V m +1 ;p +1 : TheBayesianversionoftheacceptancemodelalsowasconsideredin[7].Itwasshownthatwith aninitialdistributionthatisuniform,anoptimalpolicymaximizingtheexpectedgainistoaccept thenextballifandonlyiftheplayerhasseenatleastasmany+1ballsas )]TJ/F8 9.9626 Tf 7.748 0 Td [(1balls. 1.4. TheGeneralizedAcceptanceModel. Webeginbydeningthegeneralizedacceptanceurn. Denition1.2. Theacceptanceurninitiallycontaining m )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsand p + t "ballsiscalled the m;p ; s;t urn Thenotation,whilecomplicated,isanecessityaswewillsometimesworkwithurnsthathave dierentsetsofballweights.Next,wedenetheoutcomesofthedrawsfromtheurn,andthevalue ofthe m;p ; s;t urn. 9

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Denition1.3. A realization isasequencecontainingexactly m )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "sand p + t "s. The j th memberofthesequencereectsthe j th balldrawnoutoftheurn.Forthe m;p ; s;t urn,thereare )]TJ/F17 7.9701 Tf 5.479 -4.378 Td [(m + p p = )]TJ/F17 7.9701 Tf 5.479 -4.378 Td [(m + p m suchrealizations.Itshouldbeclearbythewaytheballsaredrawn fromtheurnthateachrealizationisequallylikely.Graphically,wecanregardeachrealization as apathfrom ;pt )]TJ/F11 9.9626 Tf 9.181 0 Td [(ms to m + p; 0usingthesteps ;s and ; )]TJ/F11 9.9626 Tf 7.749 0 Td [(t .Figure1.1showsagraphical representationoftherealization )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; +3 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; +3,fromthe ; 3;2 ; 3urn. Figure1.1. Agraphicalrepresentationoftherealization = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2,+3, )]TJ/F8 9.9626 Tf 7.749 0 Td [(2, )]TJ/F8 9.9626 Tf 7.749 0 Td [(2, +3, )]TJ/F8 9.9626 Tf 7.749 0 Td [(2,+3,fromthe ; 3;2 ; 3urn. Remark. Theballsinthe m;p ; s;t urnwiththesamevalueareconsideredtobeidentical.However, incertaininstancesitisadvantageoustoconsidersomeoralloftheballsasbeingdistinct.Whether theballshavenumberspaintedonthemornothasnobearingontheprocessinanyway. Denition1.4. Forthe m;p ; s;t urn, G m;p ; s;t isthe valueoftheurn ,thatis,theexpected gainusinganoptimalacceptancepolicy. Anoptimalacceptancepolicymaximizingtheexpectedgaincanbedescribedasafunctionofthe weight oftheurn. Denition1.5. The weight oftheurn X j isthecombinedweightoftheballsremaininginthe m;p ; s;t urnafter j ballshavebeendrawn. 10

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Inparticular,wewillalwayshave X 0 = pt )]TJ/F11 9.9626 Tf 10.414 0 Td [(ms ,and X m + p =0.Otherwise, X j isarandom variablewith X j = 8 > > < > > : X j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + s; ifthe j th balldrawnis )]TJ/F11 9.9626 Tf 7.748 0 Td [(s ," X j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(t; ifthe j th balldrawnis+ t ," when1 j m + p Theorem1.6. Anoptimalstrategymaximizingtheexpectedgain G m;p ; s;t satisesthefollowing properties: If j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ballshavebeendrawn,thentheplayershallacceptthe j th ballif X j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 > 0 If j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ballshavebeendrawn,thentheplayershallpassonthe j th ballif X j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 < 0 Proof. Weshallprovetheresultinductively.Clearly,iftheurncontainsonlyone )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ball,then weshouldnotplaceabet,otherwisewewouldlose s dollars.Iftheurncontainsonlyone+ t "ball, weshouldbetandreceivethe t dollarpayout.Giventhatweknowonoptimalstrategyoncethere arelessthan m + p ballsremainingintheurn,wedene A m;p ; s;t asthevalueifwebetonthe rstball,thenfollowanoptimalstrategythereafter; B m;p ; s;t asthevalueifwedonotbeton therstball,thenfollowanoptimalstrategythereafter;and G m;p ; s;t asthevaluefollowingan optimalstrategy.Then G m;p ; s;t =max A m;p ; s;t ;B m;p ; s;t ,and A m;p ; s;t = p m + p )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(t + G m;p )]TJ/F8 9.9626 Tf 9.963 0 Td [(1; s;t + m m + p )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F11 9.9626 Tf 7.749 0 Td [(s + G m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;p ; s;t ; B m;p ; s;t = p m + p G m;p )]TJ/F8 9.9626 Tf 9.963 0 Td [(1; s;t + m m + p G m )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;p ; s;t : Subtracting B m;p ; s;t from A m;p ; s;t ,weobtain A m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(B m;p ; s;t = pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms m + p : Anoptimalstrategyisthustoplaceabetontheballif pt>ms ,thatis,if X 0 > 0,andpasswhen pt 0,theurnis positive ,if X j < 0,theurnis negative ,andif X j =0,theurnis neutral 11

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Ifweknowthat X j = x ,wecangureouthowmanyballs a of+ t ", b of )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ofeachtype remainbysolvingthesystem a + b = m + p )]TJ/F11 9.9626 Tf 10.096 0 Td [(j;at )]TJ/F11 9.9626 Tf 10.095 0 Td [(bs = x .ObservethatTheorem1.6doesnot addresswhattheplayershoulddowhen X j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =0.Thisisbecausetheexpectedvalueofthenext balliszero,sotheplayerwillbeneitheratanadvantagenoratadisadvantageiftheplayeraccepts the j th ball.Inthiswork,weshallemploytwostrategiesthatareoptimal,whichwewillrefertoas the zero-passstrategy andthe zero-betstrategy Denition1.8. The zero-passstrategy istherulebetonthe j th ballifandonlyif X j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 > 0." Denition1.9. The zero-betstrategy istherulebetonthe j th ballifandonlyif X j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0." Inthiswork,unlessspecicallymentioned,wewillbeusingthezero-passstrategy.Forthe realizationdepictedinFigure1.1,abetwouldbeplacedoneverydrawusingthezero-betstrategy, whilewiththezero-passstrategy,abetwouldbeplacedoneverydrawexceptthethird.Generally, anoptimalstrategyisasfollows:Betwhentheurnispositive,passwhentheurnisnegative,and ipacoinwhichneednotbefairtodecidewhichoptiontousewhentheurnisneutral. Proposition1.10. Usinganyoptimalstrategyforthe m;p ; s;t urn,if s> 0 thenthelastball thatabetisplacedonisa + t ." Proof. Ifabetisplacedonthe j th ballanditisa )]TJ/F11 9.9626 Tf 7.749 0 Td [(s ",thensince X j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 0, X j = X j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + s> 0. As X m + p =0,theremustbea j +1 th ball,andabetwillbeplacedonit.Thusthe j th ballcannot bethelastballabetisplacedon. Remark .Theminimumaplayercangainusinganoptimalstrategyismax f 0 ;pt )]TJ/F11 9.9626 Tf 9.126 0 Td [(ms g .Iftheurnis initiallyunfavorable,thentheplayerwillwaituntiltheurnbecomesfavorable.Shouldthathappen, theplayerwillbetuntiltheurnbecomesunfavorableorisemptied,andwillpickupanonpositive gain.Iftheurnisinitiallyfavorable,then pt )]TJ/F11 9.9626 Tf 8.953 0 Td [(ms 0,andtheplayerwillbetuntiltheurnbecomes unfavorableorisemptied,pickingupatleast pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms 12

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2. Propertiesof G m;p ; s;t Wenowshowsomeofthepropertiespossessedbythevaluefunction G m;p ; s;t .Weshowthat foranyxed s and t ,weincrease G m;p ; s;t byaddingmore+ t "balls,removing )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls, replacing )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "ballswith+ t "balls,andaddinginballswhilekeepingtheinitialweightoftheurn xed,aswasthecasewiththeoriginalacceptanceurn.Weshowthatforxed m and p G m;p ; s;t isnondecreasingin t ,nonincreasingin s ,andcontinuousinboth s and t .Webeginwiththreebasic transformationsthathavegeometricinterpretations.Therstistheverticalstretch. Lemma2.1. Foranyreal r 0 G m;p ; rs;rt = rG m;p ; s;t Proof. Theresultwith r =0istrivial.Suppose r> 0.Denotetheweightofthe m;p ; s;t urn atstage n tobe X n andtheweightofthe m;p ; rs;rt atstage n as Y n .Withthefollowing correspondence: drawa+ rt "ballatstage n ifandonlyifa+ t "ballwasdrawn, weseethat Y n = rX n forall n and .Hence X n > 0ifandonlyif Y n > 0.Hencewebetonboth urns,orwebetonneither.Thus,thegainfromthe m;p ; rs;rt urnwillbe r timesthegainfrom the m;p ; s;t urn. Graphicallyspeaking,wetransitionfromthe m;p ; s;t urntothe m;p ; rs;rt urnbyavertical stretchwithfactor r .If t=s isrational,wecanuseLemma2.1torescaletheurntoa m;p ; s 1 ;t 1 urnwith s 1 and t 1 positiveintegerssatisfying t=s = t 1 =s 1 andgcd s 1 ;t 1 =1.Forany s and t ,the lemmaalsoallowsustouseastandardizedurncomposedof )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"and+ t=s "balls. Anotherusefultransformationisverticalreection,whichismostpowerfulinthecase s = t =1. Denition2.2. Forthe m;p ; s;t urn,the p;m ; t;s urniscalledits antiurn AnexampleoftheantiurnmapisgiveninFigure2.1.Thenexttheoremisageneralizationof Theorem2.1inChen etal. [7].Thisderivation,usingaplayertryingtomaximizegainandasecond simulatingtheantiurn,allowsustoeliminatetheabsolutevaluesignsfoundin.3. 13

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Figure2.1. Arealization fromthe ; 3;2 ; 3urnblack,andthecorresponding realizationfromthe ; 4;3 ; 2antiurnpink. Theorem2.3. TheAntiurnTheorem Foranynonnegativeintegers m and p ,andanynonnegative reals s and t ,wehave G m;p ; s;t = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms + G p;m ; t;s : Remark .Themethodofproofisonewhichwewillusetoprovemanyoftheinequalitiestofollow. Wewillsometimesaskourplayerto simulate oneurnonanother.Whatwemeanbythisisthat, givena m;p ; s;t urn,ourplayerwillbetorpassasiftheurnwereactuallya m 0 ;p 0 ; s 0 ;t 0 urn,that is,useanoptimalstrategyassociatedwiththe m 0 ;p 0 ; s 0 ;t 0 urninstead.Suchamanipulationmay includetheadditionofballswithvaluezerototheurntoaccommodatethecase m + p 6 = m 0 + p 0 Theresultisa simulationofthe m 0 ;p 0 ; s 0 ;t 0 urnonthe m;p ; s;t urn Proof. Toprovethisassertion,weshallrequiretwoplayers,whoweshallcallAdamandBetty.Adam willfollowtheoptimalzero-betstrategyassociatedwiththe m;p ; s;t urn.Bettywillsimulatethe p;m ; t;s urnonthe m;p ; s;t urninthefollowingmanner:Shewillactasifeach+ t "ballwere a )]TJ/F11 9.9626 Tf 7.749 0 Td [(t "ball,andactasifeach )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballwerea+ s "ball.Shewillfollowtheoptimalzero-pass strategyonthissimulated p;m ; t;s urn.Ifthereare p 1 + t "ballsand m 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsleft,Adam willplaceabetonthenextballifandonlyif p 1 t )]TJ/F11 9.9626 Tf 10.066 0 Td [(m 1 s 0.Betty,meanwhile,willplaceabetif andonlyif p 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(t )]TJ/F11 9.9626 Tf 8.956 0 Td [(m 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(s > 0,i.e.if p 1 t )]TJ/F11 9.9626 Tf 8.956 0 Td [(m 1 s< 0.Combined,oneandonlyoneplayerwillplace abetoneverydrawfromtheurn.ThusAdamandBetty'scombinedgainistheinitialweightof theurn, pt )]TJ/F11 9.9626 Tf 9.981 0 Td [(ms .Adam'sexpectedgainis G m;p ; s;t ,and,uponmultiplicationoftheballvalues 14

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by )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,weseethatBetty'sexpectedgainis )]TJ/F11 9.9626 Tf 7.749 0 Td [(G p;m ; t;s .Therefore,wemusthave G m;p ; s;t = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms + G p;m ; t;s : TheAntiurnTheoremwillbeaveryusefultoolforsomeoftheinequalitiesthatfollow.For inequalitiesinvolvingurnswithdierentinitialweights,thistheoremproducesasecond,reversed inequality.TheAntiurnTheoremalsoallowsustoworkexclusivelywiththeurnswith pt )]TJ/F11 9.9626 Tf 9.23 0 Td [(ms 0. Verticalreectioncanbeusedlocallythatis,onasubrealizationof when s = t ,butlocal reectionswillnotworkforthe m;p ; s;t urnsingeneral. Thethird,andperhapsmostimportant,transformationisrotation.Unlikeverticalreection, rotationcanbeappliedlocallyforallofthe m;p ; s;t urns,andinparticularitisverypowerful whenoneof s and t isanintegermultipleoftheother.Foranexample,seeGouldenandSerrano [11]. Denition2.4. Forarealization ,dene i;j ]astherealizationobtainedbyreversingtheorder the i +1 th throughthe j th ballsappearin .Inparticular,denotethe reversal R of as R = ;m + p ]. Fortherealization = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3depictedinFigure1.1,wehave ; 5]= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3 ; +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3,and ; 7]= )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; +3= .Figure2.2provides anotherexampleusingarealizationfromthe ; 3;1 ; 1urn. Lemma2.5. TheReversalLemma Wehave X k = X k )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(! i;j ] for k i and k j ,and X k )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(! i;j ] + X j + i )]TJ/F10 6.9738 Tf 6.227 0 Td [(k = X j + X i for i
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Figure2.2. Fortherealization fromthe ; 3;1 ; 1urnsolid,rotationabout thepoint P ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1producesthereversedrealization R dashed.Toobtainthe realization ; 7]dotted,onlythepartofthepath over[3 ; 7]isrotated,about thepoint Q ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1. TheresultofLemma2.5impliesthatthegraphof i;j ]canbeobtainedfromthegraphof byahalf-turnrotationofthegraphover[ i;j ]aboutthepoint P = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( i + j = 2 ; X i + X j = 2 ,or equivalently,byareectionthrough P .Thus,Lemma2.5canbeconsideredasa rotationmethod ora midpointreectionmethod ,thelatterbeingamoreappropriatetermwhenhigherdimensions areconsidered.When s =1and t isapositiveinteger,theserotationsareprimarilyimplemented viaLemma3.7,theCrossingLemma. 2.1. AHierarchyofInequalities. Wenowfocusourattentiontoboundsfortheurnfamilies, byholdingvariousparametersxed,andlettingotherparametersvary.Wendthatthevalue G m;p ; s;t increasesifweadd+ t "balls,remove )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls,addinacollectionofballswitha combinedweightofzero,orconcentratethegainsandlosses,thatis,increasing t or s while pt or ms remainsxed.Weshowtherstthreestatementsholdwiththenexttheorem. Lemma2.6. Forany m p 0 ,wehave t m + p +1 G m;p +1; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s;t t: .1 0 G m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m +1 ;p ; s;t s 1 )]TJ/F8 9.9626 Tf 30.215 6.74 Td [(1 m + p +1 : .2 t m + p +1 G m;p +1; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m +1 ;p ; s;t t + s )]TJ/F11 9.9626 Tf 30.37 6.74 Td [(s m + p +1 : .3 16

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Remark .ThisisageneralizationofTheorem2.6inChen[7].Unliketheproofthere,whichrelies ontheformulafor G m;p ;1 ; 1,wegiveacombinatorialproof. Proof. Weshowthetwoinequalities: G m;p +1; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s;t t m + p +1 ; .4 G m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m +1 ;p ; s;t 0 : .5 AnapplicationoftheAntiurnTheoremproducestheremaininginequalitiesfor.1and.2,while summing.1and.2gives.3. Toputtheurnsonequalfooting,weaddaballwithweightzerotothe m;p ; s;t urn.Theweight ofthis magicball aectsneithertheplayer'soptimalstrategy,theplayer'sgains,northeprobability ofanyrealizationeachrealizationappears m + p +1times.Therefore,anoptimalstrategywill stillresultinanaveragegainof G m;p ; s;t .Fortheotherurns,weshalldistinguishasingleball )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(a + t "forthe m;p +1; s;t urnanda )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "forthe m +1 ;p ; s;t urns thatshallcorrespondtothe magicball.Thenforallthreeurns,therewillbe )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p +1 m;p; 1 realizations.Eachrealizationfromthe m;p ; s;t urnappears m + p +1times,eachrealizationfromthe m;p +1; s;t urnappears p +1 times,andeachrealizationfromthe m +1 ;p ; s;t urnappears m +1times. Figure2.3. Identicationofballsforthe m;p +1; s;t urntop,themodied m;p ; s;t urnwithablankballcenter,andthe m +1 ;p ; s;t urnbottom. Toprove.4,wehavetheplayersimulatethe m;p ; s;t urnonthe m;p +1; s;t urnby regardingthemagic+ t "ballastheblank"ball,andfollowinganoptimal m;p ; s;t zero-bet strategybasedonthatassumption.Sincetheplayercancertainlydothis,thisisavalidstrategy, andthereforetheexpectedgaincannotexceedthevalue G m;p +1; s;t .Ifthemagicballwas infacta"ball,theplayer'sexpectedgainwouldbe G m;p ; s;t .Sincethemagicball'sweight is+ t ,"anerrorintheplayer'sfavor,theplayer'sexpectedgainwillbeatleast G m;p ; s;t .In 17

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particular,ifthemagicballisthelastdrawn,aneventoccurringwithprobability1 = m + p +1,the playerwillplaceabetsincehisstrategyiszero-betandadd t tohisgain.Therefore,theplayer's expectedgainisatleast G m;p ; s;t + t m + p +1 : Thisshows.4.Wecanshow.5bysimulatingthe m +1 ;p ; s;t urnonthe m;p ; s;t urn. Animprovementcannotbemadeonagenerallowerboundfor G m;p ; s;t )]TJ/F11 9.9626 Tf 10.341 0 Td [(G m +1 ;p ; s;t ,as G m +1 ; 0; s;t = G m; 0; s;t =0. From.4,viatheAntiurnTheorem,wehavefor m;p 0: s m + p +1 G p;m +1; t;s )]TJ/F11 9.9626 Tf 9.963 0 Td [(G p;m ; t;s s m + p +1 G m +1 ;p ; s;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(G m;p ; s;t + s G m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m +1 ;p ; s;t s 1 )]TJ/F8 9.9626 Tf 26.893 6.74 Td [(1 m + p +1 : From.5wecansimilarlyderivetheremainingboundin.1. Theassumptionthatwemustdistinguishamagicball,say,bypaintingitadierentcolorasin Figure2.3,isnotanecessity.Ifweregardtheballsasbeingidentical,thenfromthe m;p +1; s;t urnwecanhavetheplayerselectapositiveinteger i from f 1 ;:::;p +1 g uniformlyatrandom.Then, theplayercantreatthe i th + t "balldrawnasthemagicball. Ifweaddballstotheurnsothattheinitialweightremainsthesame,wewouldexpecttogainmore onaverage,simplybecausewehavemoretime,henceitwouldseemthatwehavemoreopportunity. WeshowthisisindeedthecasewithLemma2.7. Lemma2.7. Foranynonnegative m p andpositiveintegers s t G m;p ; s;t G m + t;p + s ; s;t : Remark .Withtheproperadjustments,thisresultcanbeextendedtocoverurnswith t=s rational. If t=s isirrational,wecannotaddballssothattheinitialweightremainsconstant.If t=s isrational, thevalue G m;p ; s;t mayberewrittenas C G m;p ; s 1 ;t 1 ,where C = s=s 1 = t=t 1 isanonnegative realconstantand s 1 and t 1 arepositiveintegers.Thentheextendedresultisthat G m;p ; s;t G m + t 1 ;p + s 1 ; s;t : 18

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Proof. Wewishtosimulatethe m;p ; s;t urnonthe m + t;p + s ; s;t urn.Indoingso,weshow thattheexpectedgainunderthesimulationisstill G m;p ; s;t .Thiswillrequire t + s magicballs. Weadd t + s magicballstothe m;p ; s;t urn,with s ofonecolorand t ofanothercolor.From the m + t;p + s ; s;t urn,wepaint s ofthe+ t "ballsand t ofthe )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsaccordingly,withthese ballsservingasthemagicballs.Then,forbothurns,therewillbe )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p + s + t m;p;s;t realizationsbycolor. Ourplayerwillsimulatethe m;p ; s;t urnonthe m + t;p + s ; s;t urnbyassumingeachmagic ballhasweightzero,andapplyinganoptimal m;p ; s;t bettingstrategy.Sincesuchastrategyis valid,theplayer'sexpectedgaincannotexceed G m + t;p + s ; s;t .Ontheotherhand,theexpected gainunderthesimulationisexactly G m;p ; s;t ,asgiventhattheplayerbetsandamagicballis drawn,theplayer'sexpectedgainis st )]TJ/F11 9.9626 Tf 10.004 0 Td [(ts =0,thesameasifallthemagicballshadweightzero, asinthemodied m;p ; s;t urn.Ofcourse,iftheplayerdoesnotbetnogainismade.Therefore, G m;p ; s;t G m + t;p + s ; s;t ItwouldseemconceivabletoreversetheargumentofLemma2.7.Wecansurelysimulatethe m + t;p + s ; s;t urnonthe m;p ; s;t urnwith t + s blanks,andtheexpectedgainfromthesimulation G sim m + t;p + s ; s;t isatmost G m;p ; s;t .However,becausetheblanksarelocatedwiththe simulatedurn,therealizationsfromthe m + t;p + s ; s;t urncorrespondingtorealizationsfromthe simulationdonothavethesamebettingsequence.Thus,wecannotmakethesameconclusionsas intheproof.Wecan,however,tightenthebound. Corollary2.8. Foranynonnegative m p ,andpositiveintegers s t G m;p ; s;t + st m + p + s + t G m + t;p + s ; s;t : Proof. WedothisasinthetightenedversionoftheLemma2.6.Weallowourplayertoescapethe simulated m;p ; s;t strategyonthelastdrawiftheplayerknowsitisamagic+ t "ball.Thisevent occurswithprobability s= m + p + s + t .Theplayer'sexpectedgainwiththisrevisedstrategyisthus G m;p ; s;t + st= m + p + s + t ,andasavalidstrategythiscannotexceed G m + t;p + s ; s;t Finally,weshowthatanurnwithfewerballsoflargervalue,plusorminus,ispreferabletoan urnwithmoreballsofsmallervalue.Thatis,theplayerstandstogainmorewithamorevolatile marketintheshorttermwhencomparedwithamoresteadymarketoverthelongterm.Weshall applythistooltoshowthat G m;p ; s;t isboundedif ms= pt > 1as m !1 19

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Lemma2.9. Foranyxed C 0 and m;p> 0 wehave G m;p ; s; C p G m;p +1; s; C p +1 ; and G m;p ; C m ;t G m +1 ;p ; C m +1 ;t : Proof. Forconvenience,let t = C=p and ^ t = C= p +1.Weshow G m;p ; s;t G m;p +1; s; ^ t ThesecondresultfollowsdirectlyfromtheAntiurnTheorem,asweshowbelow.Weaddamagic ballwithvaluezerotothe m;p ; s;t urn,anddesignateoneofthe+ ^ t "ballsfromthe m;p +1; s; ^ t urntocorrespondwiththisblankball. Wesimulatethe m;p +1; s; ^ t urnonthe m;p ; s;t urnbyhavingtheplayerassumethatthe + t "ballsandtheblankballareall+ ^ t "balls,usinganoptimalstrategyunderthoseassumptions. Thentheplayer'sexpectedgainisatmost G m;p ; s;t ,asthisisawell-denedstrategyforthe m;p ; s;t urn.Infact,theplayer'sexpectedgainunderthesimulationisexactly G m;p +1; s; ^ t Toshowthis,consideranarbitraryrealization fromthe m;p +1; s; ^ t urn.Since contains p +1+ ^ t "balls,andoneofthemisthemagicball,thereare p +1realizationsfromthe m;p ; s;t urnwemayassociatewith .Sincetheplayer'sstrategyisanoptimal m;p +1; s; ^ t urnstrategy, thebettingsequencesforalloftheserealizationsareidenticalandexactlythesameasthatof Supposethat i ofthe+ ^ t "ballswerebetonfor .Thenthegainfrom+ ^ t "ballsfor is i ^ t Weshowthattheaveragegainoverthecorrespondingrealizationsfromthe m;p ; s;t urnisalso i ^ t .Wehavetwocases: Case1 .Themagicballwasoneofthe i ballsbeton.Thentheactualgainsmadefromthe+ t ballsequals i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t .Thereare i suchrealizations. Case2 .Themagicballwasnotoneofthe i ballsbeton.Thentheactualgainsmadefromthe + t "ballsequals it .Thereare p +1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(i suchrealizations. Theaverageisthus 1 p +1 i i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t + p +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(i it = ipt p +1 = i ^ t: Sincethisholdsforany i andeachrealization ,weconcludethattheexpectedgainfromthe simulationis G m;p +1; s; ^ t .Thiscompletestheproof. Example .Wehave G ; 100;1 ; 1 8 : 37335,whileeachof G ; 1;1 ; 100, G ; 100;100 ; 1,and G ; 1;100 ; 100equal50.Thisalsoshowsthatwecannotmaketheinequalitiesstrict. 20

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2.2. Continuityof G m;p ; s;t in s and t Nowwefocusourattentionon s and t ,andshow thatforxed m and p G m;p ; s;t isacontinuousfunctioninboth s and t .Since G m;p ; s;t is continuousin s and t ,wehaverationalapproximationatourdisposalwhenweconsidertheurns forwhich t=s isirrational. Lemma2.10. Foranynonnegativereals s and t ,andanynonnegativeintegers m and p G m;p ; s;t isacontinuousfunctionin s and t Remark .Inprovingthisresult,weshowthemonotonicityin s and t aswell. Proof. Itsucestoshowthat G m;p ; s;t iscontinuousin t ,ascontinuityin s followsbytheAntiurn Theorem.Let > 0begiven.Weprove: G m;p ; s;t G m;p ; s;t + .6 G p;m ; t + ;s G p;m ; t;s : .7 Toshow.6,wesimulatethe m;p ; s;t urnonthe m;p ; s;t + urnbyhavingourplayerassume the+ t + "ballsare+ t "balls,andfollowinganoptimal m;p ; s;t strategy.Thentheplayer's expectedgaincannotexceed G m;p ; s;t + .Ontheotherhand,theplayer'sexpectedgainwillbe atleast G m;p ; s;t ,andstrictlysoifthereisa+ t + "ballpresent,asthatballcouldbethe lastonedrawn.Thus, G m;p ; s;t G m;p ; s;t + Weproceedsimilarlyfor.7,bysimulatingthe p;m ; t + ;s urnonthe p;m ; t;s urn.The simulationclearlygainsatmost G p;m ; t;s ,andsinceeachlossonanaccepted )]TJ/F11 9.9626 Tf 7.749 0 Td [(t "ballisless thanwhattheplayerassumes )]TJ/F11 9.9626 Tf 7.749 0 Td [(t )]TJ/F11 9.9626 Tf 8.585 0 Td [( ,thesimulationwillfetchmorethan G p;m ; t + ;s .Applying theAntiurnTheorem,weobtaintheinequality G m;p ; s;t + )]TJ/F11 9.9626 Tf 9.963 0 Td [(p G m;p ; s;t : .8 Combiningthisinequalitywith.6,wendthat G m;p ; s;t iscontinuousin t fromtheright,by letting 0.Byreplacing t with t )]TJ/F11 9.9626 Tf 9.962 0 Td [( ,weestablishcontinuityfromtheleft. Forcontinuityin s ,weapplytheAntiurnTheoremto.8,yielding G m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(m G m;p ; s + ;t G m;p ; s;t : 21

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Remark. Itshouldbenotedthattheconditionthat t and s benon-negativeisnotanecessity.If s 0and t 0,theneveryballhasanonnegativevalue,andtheplayerwillaccepteveryball, gaining pt )]TJ/F11 9.9626 Tf 9.316 0 Td [(ms .Similarly,if s 0and t 0,noneoftheballshaveapositivevalue,andtheplayer willnotacceptanyballs,andthuswillgainzero.Ifboth s and t areatmostzero,thentheplayer willwanttoacceptthe )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsinsteadofthe+ t "balls.Thus,wemultiplyboth s and t by )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 andswaptheirpositions,sothattheplayer'sexpectedgain G m;p ; s;t willequal G p;m ; )]TJ/F11 9.9626 Tf 7.749 0 Td [(t; )]TJ/F11 9.9626 Tf 7.748 0 Td [(s Clearly,overtheseranges G m;p ; s;t iscontinuous,andsincetheformsaboveholdwhen s =0or t =0,theextendedfunction G m;p ; s;t iscontinuousforall s and t 2.3. MiscellaneousResults. Here,weexaminesomeotherresultsrelatedtothe m;p ; s;t urn. Lemmas2.11and2.12relatetothemaximumweightachievedduringplay,whichweshallusewhen weexaminetheruinprobleminChapter8. Lemma2.11. Suppose m + p =^ m +^ p ,and ^ m>m .Denotetheweightofthe m;p ; s;t and ^ m; ^ p ; s;t urnsafter n ballshavebeendrawnas X n and Y n ,respectively.Thenforany ` P X n pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms + ` forsome n P Y n ^ pt )]TJ/F8 9.9626 Tf 11.846 0 Td [(^ ms + ` forsome n : Proof. First,observethattheresultistrivialwhenever ` 0,asbothprobabilitiesequalone. Therefore,suppose `> 0.Fromthe m;p ; s;t urn,wedistinguish p )]TJ/F8 9.9626 Tf 11.25 0 Td [(^ p =^ m )]TJ/F11 9.9626 Tf 10.405 0 Td [(m + t "ballsto matchupwiththedistinguished )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsfromthe^ m; ^ p ; s;t urn.Thus,eachurncontains m )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls,^ p + t "balls,and^ m )]TJ/F11 9.9626 Tf 9.962 0 Td [(m magicballs. Observethat X 0 = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms = Y 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(^ m )]TJ/F11 9.9626 Tf 9.963 0 Td [(m t + s ,andfor n 1 .9 Y n )]TJ/F11 9.9626 Tf 9.962 0 Td [(X n = 8 > > < > > : Y n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(X n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F8 9.9626 Tf 9.962 0 Td [( t + s ; ifamagicballisdrawn, Y n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(X n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; otherwise. Thus, X n Y n )]TJ/F8 9.9626 Tf 9.982 0 Td [(^ m )]TJ/F11 9.9626 Tf 9.983 0 Td [(m t + s holdsforallmatchedrealizationsandall n .Therefore,if^ isa realizationfromthe^ m; ^ p ; s;t urnwith Y n ^ ^ pt )]TJ/F8 9.9626 Tf 12.004 0 Td [(^ ms + ` ,wemusthaveforthecorresponding that X n Y n ^ )]TJ/F8 9.9626 Tf 9.963 0 Td [(^ m )]TJ/F11 9.9626 Tf 9.963 0 Td [(m t + s ^ pt )]TJ/F8 9.9626 Tf 11.846 0 Td [(^ ms + ` )]TJ/F8 9.9626 Tf 9.963 0 Td [(^ m )]TJ/F11 9.9626 Tf 9.962 0 Td [(m t + s = pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms + `: Thisprovesthelemma. 22

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TheresultofLemma2.11impliesthat P X n ` forall n P Y n >` forall n : Lemma2.12. Suppose m + p =^ m +^ p ,and ^ m>m .Thenforany ` P Y n ` forsome n P X n ` forsome n : Proof. Theresultwith ` 0istrivialsince X m + p = Y m + p =0.Assume `> 0.Notethat.9 implies X n Y n .ThenproceedingasinLemma2.11,if^ isarealizationfromthe^ m; ^ p ; s;t urn with Y n ^ ` ,then X n Y n ^ ` Lemma2.12impliesthat P X n <` forall n P Y n <` forall n ; P X n pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms + ` forsome n P Y n ^ pt )]TJ/F8 9.9626 Tf 11.846 0 Td [(^ ms + ` forsome n ; P X n >pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms + ` forall n P Y n > ^ pt )]TJ/F8 9.9626 Tf 11.846 0 Td [(^ ms + ` forall n : SimilarresultstoLemmas2.11and2.12holduponexchanging "for < "and "for > ." Next,wetakealookatsomelimitsinvolvingrst-orderdierences. Proposition2.13. Forxed m lim p !1 G m;p +1; s;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(G m;p ; s;t = t: lim p !1 G m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m +1 ;p ; s;t = s: ThisisageneralizationofTheorem4.4inChen[7],andisprovedinthesamemanner. Proof. Since,forxed p ,lim m !1 G m;p ; s;t =0,wehavebytheAntiurnTheorem: lim p !1 G m;p +1; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s;t = t +lim p !1 G p +1 ;m ; t;s )]TJ/F11 9.9626 Tf 9.962 0 Td [(G p;m ; t;s = t: 23

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Similarly, lim p !1 G m;p ; s;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(G m +1 ;p ; s;t = s +lim p !1 G p;m ; t;s )]TJ/F11 9.9626 Tf 9.963 0 Td [(G p;m +1; t;s = s: Lemma2.14. Forxed m> 0 and p> 0 ,andany : Forxed t lim s !1 G m;p ; s;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(G m;p ; s + ;t =0 : Forxed t lim s !1 G m;p ; s;t + )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s;t = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Forxed s lim t !1 G m;p ; s;t + )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s;t = p: Forxed s lim t !1 G m;p ; s;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s + ;t = m )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m X k =0 k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Proof. Weprove.Sinceweareletting s gotoinnity,assume s and s + arebothatleast pt Then .10 G m;p ; s;t = G m;p ; s + ;t = t m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Weshallprove.10withTheorem4.13.Sinceequalityholdsforsuch s ,thedierence,andhence thelimit,iszero.StatementholdsafterapplyingtheAntiurnTheoremto.Toprove,if s isgreaterthan pt and p t + ,from.10excludingthecenterexpressionwehave G m;p ; s;t + )]TJ/F11 9.9626 Tf 9.962 0 Td [(G m;p ; s;t = t + )]TJ/F11 9.9626 Tf 9.963 0 Td [(t m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : StatementfollowsfromafterapplyingtheAntiurnTheorem. Weclosethissectionwithsomeresultsrelatedtootherstrategies.Denoteby G n m;p ; s;t as theexpectedgainonthe m;p ; s;t urnifabetisplacedifandonlyiftheweightoftheurnisat least n G n m;p ; s;t astheexpectedgainifabetisplacedifandonlyiftheweightoftheurnisat 24

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most n ,andlikewisedene G >n m;p ; s;t and G 0 m;p ; s;t zero-passequal G m;p ; s;t .Ourmotivationfordeningthesestrategiesisthat somewillbeusedfortheBayesianversionoftheacceptanceurn. Lemma2.15. Forany m p s t ,and n ,thefollowinghold: G )]TJ/F10 6.9738 Tf 12.453 0 Td [(n m;p ; s;t = )]TJ/F11 9.9626 Tf 7.749 0 Td [(G n p;m ; t;s G )]TJ/F10 6.9738 Tf 12.453 0 Td [(n m;p ; s;t = )]TJ/F11 9.9626 Tf 7.749 0 Td [(G n p;m ; t;s G < )]TJ/F10 6.9738 Tf 6.226 0 Td [(n m;p ; s;t = )]TJ/F11 9.9626 Tf 7.749 0 Td [(G >n p;m ; t;s G > )]TJ/F10 6.9738 Tf 6.226 0 Td [(n m;p ; s;t = )]TJ/F11 9.9626 Tf 7.749 0 Td [(G )]TJ/F10 6.9738 Tf 6.227 0 Td [(n p;m ; t;s = pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms: Remark .Thecases n =0and n =1reducetotheAntiurnTheorem. Proof. IfAdamusesthe n -betstrategy,andBettybetsifandonlyifAdamdoesnotbet,then Adam'sexpectedgainis G n m;p ; s;t andBetty'sexpectedgainis G
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strategy,and G S m;p ; s;t denotestheexpectedgainusingstrategy S ,then G S m;p ; s;t + G S C m;p ; s;t = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms: Furthermore,if S A isthestrategycorrespondingto S viatheantiurnmap,then G S m;p ; s;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(G S C A p;m ; t;s = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms: Thislatterstatementcouldbecalledthe GeneralizedAntiurnTheorem 26

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3. The m;p ;1 ;t Urns Wenowlookatthesubcollectionofacceptanceurns,thosewith s =1and t apositiveinteger.We shallgivefourpreciseformulasfor G m;p ;1 ;t with pt m ,whichwecallthe zerocount Theorem 3.9, negativebinomial Theorem3.12, binomial Theorem3.15,and distribution Corollary3.30 forms.Afthform,the crossings formLemma4.5,willbegiveninChapter4.Fortheurnswith pt>m ,wegivethezerocountformTheorem3.25,andthedistributionformCorollary3.30, withthecrossingsformLemma4.6tofollowinChapter4.AnapplicationoftheAntiurnTheorem extendstheresultsofthischaptertotheurnswith t =1and s apositiveinteger. Theacceptanceurnswith s =1and t apositiveintegerrepresentamuchlargercollectionofurns withthepropertythatoneoftheballweightsisanintegermultipleoftheother.Theseurnsexhibit thespecialpropertydescribedbyLemma3.7,theCrossingLemma.Thispropertyfollowsbya rotation or midpointreectionmethod ,amechanismdescribedbyLemma2.5,theReversalLemma. Theoriginalacceptanceurnswith s = t arespecialintheirownright,sincebothballweightsare integermultipleseachother.Forthoseurns,wehavethemorepowerful reectionmethod atour disposal. Webeginwiththemathematicalobjectsoperatinginthebackground,thegeneralizedbinomial series.Wewillthenexaminethe m;p ;1 ; 1urns,followedbythe m;p ;1 ;t urns,andconcludethe chapterbyndingthedistributionofthegainforthe m;p ;1 ;t urns. Remark .Inthischapter,unlessotherwisestated,weshalladoptthezero-passstrategy. 3.1. GeneralizedBinomialSeries. Theexpectedgainfunction G m;p ;1 ;t forthe m;p ;1 ;t urns,particularlywhen m pt ,servesasagreatexampleofthepowerofgeneralizedbinomial series.Thegeneralizedbinomialserieswithparameter t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(see ConcreteMathematics [10] is denedas B t z = X k 0 1 tk +1 tk +1 k z k ;t real ; anditisknowntosatisfytheequation B t z =1+ z B t z t : 27

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Lagrange'sInversionformulagivesthatforanyrealnumber r .1 B t z r = X k 0 r tk + r tk + r k z k ; B t z r 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = X k 0 tk + r k z k : Areferencefortheequations.1canalsobefoundin[10,eqs..60,.61].When t =0, B 0 z =1+ z andequations.1become + z r = X k 0 r k z k ; whichisthebinomialtheorem.Inparticular, B 1 z isthegeometricseries1 = )]TJ/F11 9.9626 Tf 9.869 0 Td [(z ,while B 2 z is thegeneratingfunctionfortheCatalannumbers. Forapowerseries f ,weuse[ z n ] f z todenotethecoecientof z n in f z .From.1wecan deriveidentitiesvalidfor n 0: X k tk + r k t n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k + s n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k r tk + r = tn + r + s n ; .2 and .3 X k tk + r k t n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k + s n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k r tk + r s t n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k + s = tn + r + s n r + s tn + r + s : Thesetwoidentitiesfollowbytheidentities B t z r B t z s 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = B t z r + s 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; B t z r B t z s = B t z r + s ; andexaminingthecoecientof z n foreachexpression.Write G t;r z = X k 0 tk + r k z k = B t z r 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; and H t;r z = G t;r z G t; 0 z = B t z r )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 : Thefunction H t;r z generatesthefollowingidentity,whichshallbeveryusefultous. Lemma3.1. Foranyreal s t andinteger m 0 m X k =0 s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k kt k = m X k =0 s k t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : 28

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Proof. Notethatwemayrewrite G t; 0 z as .4 G t; 0 z = 1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 =1+ zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Since H t;r z = X k 0 tk + r k z k X k 0 tk k z k ; wehave [ z m ] H t;r z = m X k =0 t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k + r m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k tk k : Ontheotherhand,using.4,wehave H t;r z = G t;r z G t; 0 z = G t;r z + zt G t;r z B t z t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = G t;r z + zt B t z r + t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(zt B t z t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 = G t;r z + zt H t;r + t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 z : Byrepeatingtheaboveargumentiterativelyfor H t;r + t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 z ,weobtainthatforanyinteger m 0, H t;r z = m X k =0 zt k G t;r + k t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 z + zt m +1 H t;r + m +1 t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 z : Theaboveidentitygivesthat [ z m ] H t;r z = m X k =0 [ z m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ] t k G t;r + k t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 z = m X k =0 t k t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k + r + k t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = m X k =0 t k tm + r )]TJ/F11 9.9626 Tf 9.963 0 Td [(k m )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Wehavethereforeprovedthat m X k =0 t m )]TJ/F11 9.9626 Tf 9.962 0 Td [(k + r m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k tk k = m X k =0 t k tm + r )]TJ/F11 9.9626 Tf 9.963 0 Td [(k m )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Replacing r with s )]TJ/F11 9.9626 Tf 9.963 0 Td [(mt )]TJ/F8 9.9626 Tf 9.962 0 Td [(1,wehave .5 m X k =0 s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt m )]TJ/F11 9.9626 Tf 9.962 0 Td [(k kt k = m X k =0 t k s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = m X k =0 t m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k s )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 k : Itthereforeremainstoshowthat .6 m X k =0 t m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k s )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 k = m X k =0 s k t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : 29

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Weshallusethefollowingidentity )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(see[10,eq..19] : m X k =0 r + k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 k x k x + y m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k = m X k =0 m + r k x k y m )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ; integer m: Bytaking x =1, y = t )]TJ/F8 9.9626 Tf 9.86 0 Td [(1and r = s )]TJ/F11 9.9626 Tf 9.86 0 Td [(m ,equation.6nowfollowsandourproofofthetheorem iscomplete. Lemma3.1linksthezerocount,negativebinomial,andbinomialformsof G m;p ;1 ;t .Ofthese, thebinomialformistheeasiesttoanalyzewhen m or p islarge.Thisisbecausewecanrewritethe right-handsideof.6intermsofabinomialdistribution. 3.2. The m;p ;1 ; 1 Urns. Beforeweexaminethe m;p ;1 ;t urns,westartwiththeoriginal m;p ;1 ; 1urns. Theorem3.2. Chen etal Forany m and p G m;p ;1 ; 1=max f 0 ;p )]TJ/F11 9.9626 Tf 9.962 0 Td [(m g + m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 min f m;p g)]TJ/F7 6.9738 Tf 10.308 0 Td [(1 X k =0 m + p k : Theorem3.2givesthe distributionform of G m;p ;1 ; 1.Weshallndthedistributionofthegain ofthe m;p ;1 ; 1urnswithanoptimalstrategyviathe reectionmethod .Recallthatwereecta realization byswappingthesignsoftheballs,exchangingplusforminusandviceversa.This willreect throughthe n -axis,resultinginarealization^ fromtheantiurn.Here,weshallapply reectionlocally,tosubrealizationswithin .Graphically,afteranyreectionsareperformed,the variousdisjointsubpathsreectedornotareshiftedverticallysothatacontinuouspath 0 ending atthepoint m + p; 0isformed. Toobtainthedistributionofthegain,wewillreectportionsoftherealizationsthatresultina netgainofone. Denition3.3. If X n = a forsome n X q = b forsome q>n ,and X k 6 = b for n
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Figure3.1. Forthisrealizationfromthe ; 8;1 ; 1urnblack,thedashedportionisa )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripfromneutral,andresultsinagainofonewhenthezero-bet strategyisused.FortheproofofLemma3.4with k =1,this )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripisreected, resultinginthepathfromthe ; 7;1 ; 1urn )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(pinkover[0 ; 11,blackover[11 ; 18] Theblackrealizationgains2usingthezero-betstrategy,asthereisasecond )]TJ/F8 9.9626 Tf 7.749 0 Td [(1" tripfromneutralover[14 ; 17]. Lemma3.4. Foranyintegers m p ,integer min f 0 ;p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m g k p ,andanyoptimalstrategy, P Playergainsatleast k = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Proof. Weassumerstthatplayerusesthezero-betstrategy.Supposethat m p .Weshallcount thenumberofrealizationsforwhichtheplayergainsatleast k .Notethatwhentheurnrstreturns toneutral,theplayerstartsbettinguntiltheurnreachesaweightof )]TJ/F8 9.9626 Tf 7.748 0 Td [(1,withtheplayergaining one.Theplayerthenstopsbettinguntiltheurnnextbecomesneutral.Thismustoccuratleast k timesinorderfortheplayertogainatleast k .Let bearealizationgainingatleast k .Eachtime theplayergainsone,theurnwouldhavetomakea )]TJ/F8 9.9626 Tf 7.748 0 Td [(1"tripfromneutral,followedbya+1"trip backtoneutral. Weshallshowthatthereisone-to-onecorrespondencebetweentherealizations thatgainat least k fromthe m;p ;1 ; 1urnandtherealizations 0 fromthe m + k;p )]TJ/F11 9.9626 Tf 10.003 0 Td [(k ;1 ; 1urn.Toturn into 0 ,wesimplyreecttherst k )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripsof fromneutral.Notethatthenumberof+1" ballsinvolvedineach )]TJ/F8 9.9626 Tf 7.748 0 Td [(1"tripisonemorethanthenumberof )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balls.Thus,eachreectionof a )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"triphastheneteectofincreasing m byoneanddecreasing p byone.Hence,byreecting therst k )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripsfromneutral,wehaveturned into 0 .Togetback from 0 ,wenoterst thattheinitialurnweightof 0 is p )]TJ/F11 9.9626 Tf 8.797 0 Td [(m )]TJ/F8 9.9626 Tf 8.797 0 Td [(2 k 0,andthus 0 containsa+ m )]TJ/F11 9.9626 Tf 8.797 0 Td [(p +2 k "trip,which wepartitionintoa+ m )]TJ/F11 9.9626 Tf 9.826 0 Td [(p "trip,and2 k +1"trips.Startingwiththerstofthose+1"trips, wereecteveryother+1"trip,creating k segmentscomposedofa )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripfollowedbya+1" 31

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trip.Sincetheyfollowa+ m )]TJ/F11 9.9626 Tf 10.031 0 Td [(p "triptoneutral,thenow)]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripsarefromneutral,andthe +1"tripsarebacktoneutral.Thus,wehaveshownthatthemapisaone-to-onecorrespondence. Sincethenumberofrealizationsofa m + k;p )]TJ/F11 9.9626 Tf 10.087 0 Td [(k ;1 ; 1urnequals )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p )]TJ/F17 7.9701 Tf 6.587 0 Td [(k ,wehavethusshown thatfor m p andthezero-betstrategy, m + p p P Playergainsatleast k = m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Forthezero-passstrategy,inorderfortheplayertogainone,theweightoftheurnhastorst reach+1,andtheplayerwillstartbettinguntiltheurnnextreturnstoneutral.Thereectionwill beoftherst k )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"tripsfrom1.Theproofissimilar,andthedetailsareomitted.Anyother optimalstrategyisaprobabilistichybridofthezero-betandzero-passstrategies.Therefore,the resultholdstruehereaswell. Finally,theresultwith p>m followsbyreectionofthe )]TJ/F8 9.9626 Tf 7.748 0 Td [( p )]TJ/F11 9.9626 Tf 9.956 0 Td [(m "tripfromthestart.Then, againofatleast k p )]TJ/F11 9.9626 Tf 10.106 0 Td [(m fromthe m;p ;1 ; 1urncorrespondstoagainofatleast k )]TJ/F8 9.9626 Tf 10.106 0 Td [( p )]TJ/F11 9.9626 Tf 10.106 0 Td [(m fromthe p;m ;1 ; 1urn.Thus, P Playergainsatleast k = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p m )]TJ/F1 9.9626 Tf 9.963 8.069 Td [()]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : FromLemma3.4,wehavefor m p G m;p ;1 ; 1= p X k =1 P playergainsatleast k = m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 m + p k ; afteranindexshift.Thecasefor p>m isobtainedsimilarly. Theprobabilityaplayergainsexactly k usinganoptimalstrategyequalstheprobabilitythatthe maximumweighttheurntakesduringplayequals k .Thisisnocoincidence;wewillshowthatthis alsoholdswhen s and t arearbitrarynonnegativerealsinChapter4,withTheorem4.18.Fornow, weshowthatthisholdswhen s = t =1,butweshallgiveadierentpresentationoftheresult. Lemma3.5. If m p ,then P X n k forsome n = P X n = k forsome n = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : If p m and k 0 isaninteger,then P X n p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + k forsome n = P X n = p )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + k forsome n = m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 m + p p + k : 32

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Proof. Suppose m p .Let bearealizationfromthe m;p ;1 ; 1urnwith X j = k and j minimal.Observethattherst j ballsforma+ m )]TJ/F11 9.9626 Tf 10.217 0 Td [(p + k "trip,andtheremaining m + p )]TJ/F11 9.9626 Tf 10.218 0 Td [(j ballscontain k more+1"sthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"s.Thus,byreectingthelast m + p )]TJ/F11 9.9626 Tf 10.335 0 Td [(j ballsweobtaina realization^ fromthe m + k;p )]TJ/F11 9.9626 Tf 10.059 0 Td [(k ;1 ; 1urn.Thismappingisreversible,aseachrealizationfrom the m + k;p )]TJ/F11 9.9626 Tf 9.489 0 Td [(k ;1 ; 1urnmuststartwitha+ m )]TJ/F11 9.9626 Tf 9.488 0 Td [(p + k "trip,since m p .Itfollowsthatthere are )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k realizationsthatreachtheweight k duringplay. Theresultwith p m issimilar.Thistime,wereectthe+ k "tripfromtheinitialweight p )]TJ/F11 9.9626 Tf 8.897 0 Td [(m instead.Weomittheremainingdetails. When s = t ,reectionisaveryusefultool,asweshallseewhenweexaminetheBayesian versionoftheurninChapter7andtheruinprobleminChapter8.When s and t arearbitrary, theonlyusefulapplicationofthereectionmethodistheantiurnmap.However,when s =1and t isanintegerortheantiurn/verticalstretchequivalent,wehaveanothergeometrictool,rotation reversal,ormidpointreection,thatwecanuse. 3.3. TheValueofthe m;p ;1 ;t Urn. Forthe m;p ;1 ;t urn,therearetwowaystomakewhat weshallcalla permanentgain .Therstisoutlinedbelow. Suppose X n =0and n 6 = m + p .Dene = n =min f h>n : X h 0 g ;G n = 8 > > < > > : X n +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(X ; if X n +1 =1, 0 ; otherwise. G n isthusthegainincurredfromtheseriesofbetsonballs n +2throughball n .Notethatsince wehave )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balls,iftheplayerdoesnotbetonthe n th ballandbetsonthe n +1 th ball,then wemusthave X n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =0and X n =1. Theonlyotherwayagaincanbemadeisifabetisplacedonthe rst ball.Then X 0 = pt )]TJ/F11 9.9626 Tf 8.163 0 Td [(m> 0. When pt>m ,theplayerbetsuntiltheurnbecomesnonpositive,pickingupbetween pt )]TJ/F11 9.9626 Tf 10.23 0 Td [(m and pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1.Todealwiththisambiguity,wedenefor pt>m .7 =min f n : X n 0 g ; = E [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(X ] : If pt m ,wedene tobezero.Wecall the crossingnumber .Clearly,wehavethat0 t )]TJ/F8 9.9626 Tf 8.618 0 Td [(1. Infact,weshallsoonshowthat0 < t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = 2if pt>m> 0. 33

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Theorem3.6. Forany m and p G m;p ;1 ;t =max f 0 ;pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m g + + X n 6 = m + p E G n j X n =0 P X n =0 : Proof. Inthecasewhentheurnisinitiallypositive,theplayerwillbetuntilthersttimetheurnis nonpositive,averagingagainof pt )]TJ/F11 9.9626 Tf 9.372 0 Td [(m + .Afterthat,theplayer,usingthezero-passstrategy,will notplaceabetuntilthersttime,saytime n ,whentheweightoftheurnispositive.Theplayer willgain G n untilthenexttime X 0.Theprocessisthenrepeateduntiltheurnisempty.Thus, ifwelet n = 8 > > < > > : 1 ; if X n =0, 0 ; otherwise, thegainoftheplayeris P n 6 = m + p G n n ,andhencethevalueis G m;p ;1 ;t =max f 0 ;pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m g + + X n 6 = m + p E G n n =max f 0 ;pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m g + + X n 6 = m + p E G n j X n =0 P X n =0 : Wefocusnowonthetwoobjectsinthesum, P X n =0and E G n j X n =0.If X n =0,then foreach+ t "ballpresent t )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ballsmustbepresent.Hencetheremustbe kt + k ballsleftinthe urnwith k p kt m after n ballshavebeendrawn.Sinceeachrealizationisequallylikely, P X n =0= kt + k k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;n = m + p )]TJ/F11 9.9626 Tf 8.078 0 Td [(k )]TJ/F11 9.9626 Tf 8.079 0 Td [(kt; 0 k min f p; b m=t cg : Asfor E G n j X n =0,webeginwithaveryimportantlemmathatwillallowustogivealternate formsoftheexpectedgainwhen m pt Lemma3.7. TheCrossingLemma Forthe m;p ;1 ;t urn,let S := f : X n = X q =0 ;X k 6 =0 for n
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Remark .Wehaveset X n =0= X q forconvenience. Proof. Let 2 S .Theunionoftheevents f A i g t i =0 is A := X k 0, X k +1 0forsome n kt )]TJ/F11 9.9626 Tf 10.022 0 Td [(i ,therefore ;q ]isalsoin S .Furthermore,thelasttimethe urntakestheweight t )]TJ/F11 9.9626 Tf 9.26 0 Td [(i in ;q ]beforetime q isafterthe th ballhasbeendrawn.Therefore,we caninvertthemapandrecover from ;q ],sothemapisinjective.Given 0 in S withtheevent A 0 ,sincetheurnweightincreasesbyones,theurnweightmustbe t )]TJ/F11 9.9626 Tf 10.19 0 Td [(i atsomepointbetween n and q .Therefore,themapissurjectiveaswell.Thus, j A i j = j A 0 j forall i Since S canbepartitionedintotheequal-sizeclasses A i ,0 i t ,weconcludethat P A i j 2 S = 1 t +1 ; asdesired. Corollary3.8. E G n j X n =0= t 2 : Proof. Suppose X n =0.Fixthenexttimetheurnisneutral,after q ballshavebeendrawn.If G n = i ,where1 i t ,thentheevent A i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 hasoccurredon n;q ],andif G n =0, A t hasoccurred on n;q ].Sinceeach A i isequallylikely, E G n j X n = X q =0= 1 t +1 t X i =0 i = t 2 : Sincethisholdsforanysuch q>n ,weconcludethat E G n j X n =0= t 2 : 35

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Wenowgiveexplicitformulasfor G m;p ;1 ;t when m pt .Weshallexaminetheurnswith m
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Lemma3.10. Suppose m pt .Thenforinteger k 0 wehave P N k = t +1 k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Proof. Assume N k .Weshallassumeinitiallythattheevent A t followsforeachofthenal k timestheurnisnonemptyandneutral.Ifthereare M realizationssatisfyingthisproperty,then theCrossingLemmaimpliesthatthenumberofrealizationsforwhich N k is M t +1 k ,since wehave t +1equallylikelyoptionsfortheevent A i tofolloweachtimetheurnisnonemptyand neutral,independentofeachother. Forarealization with N k and A t followingthelast k timestheurnisnonemptyandneutral, wemaywrite as = TM k Q k M 1 Q 1 ; where M i Q i M 1 Q 1 isthe i th smallestnonemptysuxof thathasweightzero )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(thatis,theurn isneutralforthe i th tolasttimewith j M i Q i M 1 Q 1 j ballsleft ,and M i isasolitary+ t "ballfor each i .Therefore, M i isanindicatoroftheevent A t ,andsince M i Q i hasweightzero, Q i hasweight )]TJ/F11 9.9626 Tf 7.749 0 Td [(t foreach i .Furthermore,noproperprexof Q i canhaveweight )]TJ/F11 9.9626 Tf 7.749 0 Td [(t byourchoiceof M i Q i .The identicationofthesesegmentswithinarealizationfromthe ; 4;1 ; 2urnforthecase k =2is showninFigure3.2. Figure3.2. Segmentidenticationofarealizationfromthe ; 4;1 ; 2urnwith N 2.UsingtheCrossingLemmaover M 1 Q 1 and M 2 Q 2 ,thisrealizationis associatedwith3 2 =9realizationsfromthe ; 4;1 ; 2urnwith N 2. Wemap totherealization 0 fromthe m;p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ;1 ;t urn,where 0 = Q k Q k )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Q 1 T: 37

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Fromarealization 0 fromthe m;p )]TJ/F11 9.9626 Tf 10.948 0 Td [(k ;1 ;t urn,werecoverauniquerealization fromthe m;p ;1 ;t urnwiththedesiredpropertiesasfollows:Sincetheurnweightincreasesbyonesand m pt ,wecanndthesmallestprexof 0 withweight )]TJ/F11 9.9626 Tf 7.749 0 Td [(it ,1 i k .Thisshallbe Q k Q k +1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(i Thus,wecanrecover Q i foreach i .Wethenmovetheword Q k Q 1 totheendoftherealization, andinserta+ t "ballbeforeeach Q i .Therefore,thenumberofrealizations M equals )]TJ/F17 7.9701 Tf 5.479 -4.378 Td [(m + p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k p )]TJ/F17 7.9701 Tf 6.587 0 Td [(k thenumberofrealizationsforwhich N k equals )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p )]TJ/F17 7.9701 Tf 6.587 0 Td [(k p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k t +1 k ,thus P N k = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t +1 k ; asdesired. Corollary3.11. Foranyinteger k 0 P N = k = t +1 k m )]TJ/F8 9.9626 Tf 9.962 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Remark .Theresultisobtainedbytakingthedierence P N = k )]TJ/F11 9.9626 Tf 9.657 0 Td [(P N = k +1.Thisresultcan alsobeshowncombinatorially.Therealizationswith N =0are ballotpermutations .Intermsofour terminologythusfar,anentirerealizationisa+ m )]TJ/F11 9.9626 Tf 9.808 0 Td [(pt "tripifandonlyif N =0.Thesewillbe discussedingreaterdetailinChapter6. Theorem3.12. NegativeBinomialForm Suppose m pt .Then G m;p ;1 ;t = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 m + k k t +1 p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k : Proof. Wehave G m;p ;1 ;t = t 2 E N = t 2 X k 1 P N k ; afterwhichwereindex,summingover p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k Wecallthisformof G m;p ;1 ;t thenegativebinomialformbecauseofthepresenceofthebinomial coecient )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + k k .Withafewadjustments,wecanassociatethisformwithanegativebinomial distribution. Wenowworktowardthe binomialform ,usingtherandomvariable N + = f n : X n =0 ;X n +1 =1 g : 38

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Lemma3.13. If m pt ,then P N + k = t k m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : Proof. Werstassumethatthelast k occurrencesoftheevent N + arefollowedbytheevent A 0 .Then,ifthereare M realizationswiththisproperty,bytheCrossingLemmathereare Mt k realizationswith N + k .Notetheevent A t isineligible,asthatisthecontributorto N )]TJ/F8 9.9626 Tf 6.725 -3.616 Td [(.We showthat M = )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p )]TJ/F17 7.9701 Tf 6.587 0 Td [(k Wemaywritearealization with N + k as = TP k M k Q k P k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 M k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P 1 M 1 Q 1 ; where M i isasequenceconsistingofalone+ t "ball, P i hasweight )]TJ/F11 9.9626 Tf 7.749 0 Td [(t foreach i ,andeachproperprexof P i hasnegativeweight,and Q i hasweightzeroforeach i ,andeachprexof Q i hasnonpositiveweight. Theidenticationofthesesegmentsinarealizationfromthe ; 4;1 ; 2urnwith N + 2isgiven inFigure3.3.Notethat j Q i j maybezero,theinitial )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ballatthebeginningof P i contributes onetoward N + thus N + k ,while M i servesasanindicatoroftheevent A 0 .Wemap tothe followingrealizationfromthe m + k;p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ;1 ;t urn: 0 = Q 1 M 0 1 Q k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 M 0 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q k M 0 k TP k P k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P 1 ; where M 0 i denotesasingle )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ball. Wenowshowthatthemapisreversible.Let 0 bearealizationfromthe m + k;p )]TJ/F11 9.9626 Tf 9.677 0 Td [(k ;1 ;t urn. Thesmallestprexof 0 withweight )]TJ/F11 9.9626 Tf 7.748 0 Td [(i ,1 i k ,wesetas Q 1 M 0 1 Q i M 0 i notingthatthelast memberofthisprexmustbea )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ball.Thesmallestsuxof withweight )]TJ/F11 9.9626 Tf 7.749 0 Td [(it ,1 i k ,is thesequence P i P i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P 1 .Since m pt ,wehave pt )]TJ/F11 9.9626 Tf 8.544 0 Td [(m )]TJ/F11 9.9626 Tf 8.544 0 Td [(k t +1 )]TJ/F11 9.9626 Tf 18.265 0 Td [(k t +1,thus Q 1 M 0 1 Q k M 0 k and P i P i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 P 1 donotintersect.Therefore,wecanreversethemap,changing M 0 i backto M i Wehavethusshownthat M = )]TJ/F17 7.9701 Tf 5.48 -4.378 Td [(m + p p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k .ItfollowsfromtheCrossingLemmathatthenumberof realizationswith N + k equals t k )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p )]TJ/F17 7.9701 Tf 6.587 0 Td [(k ,andthus P N + k = t k m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : 39

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Figure3.3. Segmentidenticationofarealizationfromthe ; 4;1 ; 2urnwith N + 2.UsingtheCrossingLemmaonthesegments P 1 M 1 and P 2 M 2 ,thisrealizationisassociatedwith2 2 =4realizationsfromthe ; 4;1 ; 2urnwith N + 2. Observethat Q 1 isempty. Corollary3.14. Foranyinteger k 0 ,wehave P N + = k = t k m + k +1 )]TJ/F8 9.9626 Tf 9.962 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t m + k +1 m + p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Remark .Therealizationswith N + =0are weakballotpermutations .Weshalluseanalternative name,andcalltherealizationswith N + =0 zero-gainrealizations ,assuchrealizationswillgive aplayeragainofzeroifthezero-passstrategyisused.Ifweadjoina )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balltotheendofa realization with N + =0,thentheextendedrealization 0 wouldbea+ m )]TJ/F11 9.9626 Tf 8.909 0 Td [(pt +1"trip.Again, seeChapter6formoredetails. Theorem3.15. BinomialForm Suppose m pt .Then G m;p ;1 ;t = )]TJ/F11 9.9626 Tf 9.637 6.739 Td [(t 2 + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p X k =0 m + p +1 k t p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Proof. Wehave E N = t +1 t E N + = t +1 t m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 m + p k t p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k ; andthissumequals m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 m + p k t p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k + p X k =1 m + p k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 t p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k # = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 m + p +1 k t p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k ; afterwhichwemultiplythroughby t= 2. 40

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Thetransformationofthesummationof G m;p ;1 ;t fromthezerocountform,throughthe negativebinomialform,andtothebinomialformcanalsobedoneviaLemma3.1,withinthat result m replacedby p t replacedby t +1,and s replacedby m + p +1: .8 p X k =0 kt + k k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k = p X k =0 m + k k t +1 p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k = p X k =0 m + p +1 k t p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Forthethirdindicator, N )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(,themethodthatproducedthedistributionof N + fails.Thisis becauseeachoccurrenceof N + comeswithafactorof t ,andwewillhavenowaytokeeptrackof howmanytimes N + occursusingthemethodthatgavetheresultofLemma3.13.Wecanobtain thedistributionof N )]TJ/F8 9.9626 Tf 9.493 -3.615 Td [(= f n : X n =0 ;X n +1 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(t g usinganothermethod. Lemma3.16. Suppose m pt .Then,forany 0 i t P A i occurs k times = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X j = k m )]TJ/F8 9.9626 Tf 9.963 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j j k t j )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Inparticular, P A t occurs k times = P N )]TJ/F8 9.9626 Tf 9.492 -3.616 Td [(= k : Proof. Corollary3.11impliesthatfor0 j p P N = j = t +1 j m )]TJ/F8 9.9626 Tf 9.963 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Therefore,givenaspecicsequenceofcrossingevents A x 1 ;:::;A x j ,thereare m )]TJ/F8 9.9626 Tf 9.962 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j realizationswith N = j andthatparticularcrossingsequence.Given i ,ourtaskistocountthe numberofsequencesforwhich A i occurs k times.Clearly,wemusthave k j p ,andthere are )]TJ/F17 7.9701 Tf 5.849 -4.379 Td [(j k t j )]TJ/F10 6.9738 Tf 6.227 0 Td [(k sequencesforwhich A i appears k times.Wethensumover j anddivideby )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p to completetheproof. Lemma3.17. Suppose m pt .Then,forany 0 i t P A i happensatleast k times = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X j = k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t j )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Proof. TheproofissimilartotheproofofLemma3.16inmanyrespects.Toavoiddouble-counting, werequirethattherstcrossingeventweactuallycountthatis,the j th fromlastcrossingevent isindeed A i 41

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Wecanusethesesameprocedurestoobtain N + from N .Combinatorialidentitiesresultwhen comparedwiththeresultsinLemmas3.10and3.13.Westatethefollowingresultsherewithout proof. Lemma3.18. Suppose m pt .Then,forany 0 i t P N + = k = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X j = k m )]TJ/F8 9.9626 Tf 9.963 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j j k t k ; and P N + k = m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p X j = k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(j j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 t k : Wecanalsouseourresultsabout N + toobtaininformationon N )]TJ/F8 9.9626 Tf 6.724 -3.615 Td [(. Lemma3.19. Suppose m pt .Then,forany 0 i t P A i occurs k times = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X j = k m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + j t +1+1 m + j +1 m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(j j k t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 j )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ; and P A i occursatleast k times = m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p X j = k m + p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 j )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Proof. TheprooffollowssimilarlytotheproofofLemma3.16,when0 i t )]TJ/F8 9.9626 Tf 10.135 0 Td [(1.Notethatthe event A t issegregatedfromtheothercrossingevents.Forthecase i = t i.e.thedistributionof N )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(,wersttakethejustproven i =0case,thenapplytheCrossingLemmabydoingacircular shifton f 0 ;:::;t g ,sending i to i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1modulo t +1. 3.3.2. OtherStrategies. Sincetheurnweightcanonlyincreasebyones,wecangiveexplicitformulas forsomeofthealternatestrategiesgiveninsection2.3.Somewillbeusedfortherandomversion oftheurn,whichwediscussinChapter7. Theorem3.20. Suppose m pt .Thenfor pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m r 0 G r m;p ;1 ;t = r + G m;p ;1 ;t : Proof. Weshowthatforeach k ,theprobabilityofgaining r + k withthe r -betstrategyisthesame astheprobabilityofgaining k withthezero-betstrategy.Since m pt ,eachrealization begins witha+ m )]TJ/F11 9.9626 Tf 10.096 0 Td [(pt "tripfromtheinitialweight pt )]TJ/F11 9.9626 Tf 10.097 0 Td [(m .Sincetheurnweightcanonlyincreaseby 42

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ones,wecanbreakthistripintoa )]TJ/F11 9.9626 Tf 7.749 0 Td [(r "tripfrom pt )]TJ/F11 9.9626 Tf 9.018 0 Td [(m anda+ m )]TJ/F11 9.9626 Tf 9.017 0 Td [(pt + r "tripfrom pt )]TJ/F11 9.9626 Tf 9.017 0 Td [(m )]TJ/F11 9.9626 Tf 9.017 0 Td [(r Writing = QT ,with Q theinitial )]TJ/F11 9.9626 Tf 7.749 0 Td [(r "trip,if gains k withthezero-betstrategy,thenthe realization 0 = TQ R willgain k + r withthealternatestrategy,asthebettingsequencesover T forboth and 0 areidentical,whilenobettingoccursduring Q in whilenopassingoccursover Q R in 0 .Thus,theplayerwillgain r moreon 0 comparedwith .Theinversemapcanbefound byndingthelasttimetheurnweightis r ,reversingtheballsthatfollow,andshiftingthemtothe beginningoftherealization.Thisresultholdsoverallrealizations ,completingtheproof. TheExtendedAntiurnTheoremthenimpliesthefollowingCorollary. Corollary3.21. Suppose m pt .Thenfor pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m r 0 G r m;p ;1 ;t = G r +1 m;p ;1 ;t ,andwehavethefollowing. Corollary3.22. Suppose m pt .Thenfor pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 r )]TJ/F8 9.9626 Tf 18.265 0 Td [(1 G >r m;p ;1 ;t = r +1+ G m;p ;1 ;t ; and G r m;p ;1 ;t = pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m )]TJ/F11 9.9626 Tf 9.963 0 Td [(r )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(G m;p ;1 ;t : 3.3.3. The m;p ;1 ;t Urnswith mm ,then P X = )]TJ/F11 9.9626 Tf 7.749 0 Td [(t +1 P X = )]TJ/F11 9.9626 Tf 7.749 0 Td [(t +2 P X =0 : Proof. Considerarealization with X = )]TJ/F11 9.9626 Tf 7.749 0 Td [(i i> 0.Wewillprovideaninjectivemapofsuch realizationsintorealizationswith X = )]TJ/F11 9.9626 Tf 7.749 0 Td [(i +1.Asketchofthemappingwewilluseisdepictedin Figure3.4. 43

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Figure3.4. Arealizationwithcrossingnumber i> 0,andtherealizationwith crossingnumber i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1associatedwithit. With X = )]TJ/F11 9.9626 Tf 7.748 0 Td [(i ,wehave X )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = t )]TJ/F11 9.9626 Tf 10.022 0 Td [(i .Sincewehave )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ballsand X m + p =0,theurnwill haveweight )]TJ/F11 9.9626 Tf 7.748 0 Td [(i +1atsomestage.Let q denotethesmallestvaluesuchthat X q = )]TJ/F11 9.9626 Tf 7.748 0 Td [(i +1.We willexaminetherealization )]TJ/F8 9.9626 Tf 9.619 0 Td [(1 ;q ].Clearly, X j )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(! )]TJ/F8 9.9626 Tf 9.619 0 Td [(1 ;q ] > 0while0 j )]TJ/F8 9.9626 Tf 9.62 0 Td [(1.Asforthe range j q )]TJ/F8 9.9626 Tf 9.962 0 Td [(1,wehavebytheReversalLemmathat X q + )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(! )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;q ] + X j = X )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + X q = t )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 i +1 : Fortherange j q )]TJ/F8 9.9626 Tf 8.189 0 Td [(1wemusthave X j )]TJ/F11 9.9626 Tf 18.265 0 Td [(i ,andthuswehave X j )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(! )]TJ/F8 9.9626 Tf 8.189 0 Td [(1 ;q ] t +1 )]TJ/F11 9.9626 Tf 8.189 0 Td [(i> 0 inthesamerange.Therefore, q isthersttimethat X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(! )]TJ/F8 9.9626 Tf 8.546 0 Td [(1 ;q ] isnegative.Themapisinjective aswell-thefactthat X j )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(! )]TJ/F8 9.9626 Tf 10.187 0 Td [(1 ;q ] t +1 )]TJ/F11 9.9626 Tf 10.186 0 Td [(i> 0for j q )]TJ/F8 9.9626 Tf 10.187 0 Td [(1impliesthatifmorethan realizationismappedinto )]TJ/F8 9.9626 Tf 10.694 0 Td [(1 ;q ],thenboth and q arethesameforboth.Sincenothing changesbeforestage )]TJ/F8 9.9626 Tf 9.369 0 Td [(1andafterstage q ,andthereversalisinvertiblegiven )]TJ/F8 9.9626 Tf 9.369 0 Td [(1and q ,andthe tworealizationsmustbeidentical. If0 t )]TJ/F11 9.9626 Tf 10.175 0 Td [(r ,weneedtodiscardtherealizationsfromthe pt;p ;1 ;t urnthatdonotstartwith r )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balls.Wecancountthesebadrealizationseasily,since 44

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theserealizationsbeginwith t )]TJ/F11 9.9626 Tf 9.585 0 Td [(i > < > > : 1 t +1 pt + p p ; 0 i t )]TJ/F11 9.9626 Tf 9.962 0 Td [(r 1 t +1 pt + p p )]TJ/F1 9.9626 Tf 9.962 14.047 Td [( p )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 t +1+ i p )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(r +1 i t )]TJ/F8 9.9626 Tf 9.962 0 Td [(1. Inparticular,if r =1,theneachcrossingisequallylikely,andthus = t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = 2. Forothercases,wecanatleastuseLemma3.23tolimittherangeof Lemma3.24. If pt>m and m> 0 ,then 0 < t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = 2 Proof. Denote P X = )]TJ/F11 9.9626 Tf 7.748 0 Td [(i as p i ,for0 i t )]TJ/F8 9.9626 Tf 9.993 0 Td [(1.Lemma3.23tellsusthat p t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p 1 p 0 : Forany i inthisrange, ip i + t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(i p t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(i = p i + p t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(i t )]TJ/F8 9.9626 Tf 8.855 0 Td [(1 2 + p t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(i )]TJ/F11 9.9626 Tf 9.962 0 Td [(p i t )]TJ/F8 9.9626 Tf 8.856 0 Td [(1 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(i p i + p t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(i t )]TJ/F8 9.9626 Tf 8.855 0 Td [(1 2 : Thus = t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 ip i t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 p i = t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 ; withequalityholdingifandonlyif p 0 = p 1 = = p t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Asfortheremovaloftheequalsignon thelowerbound,therealization 0 consistingofdrawingallbutoneofthe )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balls,followedby allofthe+ t "balls,followedbythelast )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ballhas X 0 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1.Sincetherearenitelymany realizations,wemusthave > 0. Wecannowpresentthe zerocount formfor m
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Remark .SumsforwhichtheupperlimitislessthanthelowerlimitlikeTheorem3.25with m > > < > > > : m + p p ; if ` pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + p d `=t e X k =0 m )]TJ/F11 9.9626 Tf 8.856 0 Td [(pt + ` m )]TJ/F11 9.9626 Tf 8.855 0 Td [(pt + kt + k + ` m )]TJ/F11 9.9626 Tf 8.856 0 Td [(pt + kt + k + ` k pt + p )]TJ/F11 9.9626 Tf 8.856 0 Td [(k )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt )]TJ/F11 9.9626 Tf 8.856 0 Td [(` p )]TJ/F11 9.9626 Tf 8.855 0 Td [(k ; otherwise. Proof. Sinceboththeweightszeroand pt )]TJ/F11 9.9626 Tf 10.47 0 Td [(m arealwaystaken,theresultwhen ` pt )]TJ/F11 9.9626 Tf 10.47 0 Td [(m + istrivial.Suppose `> pt )]TJ/F11 9.9626 Tf 10.467 0 Td [(m + .Thenarealizationreaching ` beginswitha+ m )]TJ/F11 9.9626 Tf 10.467 0 Td [(pt + ` trip.Fromthere,anymethodofreturningtozerowilldo.Thereare m )]TJ/F17 7.9701 Tf 6.587 0 Td [(pt + ` m )]TJ/F17 7.9701 Tf 6.587 0 Td [(pt + kt + k + ` )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m )]TJ/F17 7.9701 Tf 6.587 0 Td [(pt + kt + k + ` k such+ m )]TJ/F11 9.9626 Tf 9.625 0 Td [(pt + ` "tripscontaining k + t "balls,asaresultofCorollary3.11,andthusthereare )]TJ/F17 7.9701 Tf 5.479 -4.378 Td [(pt + p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k )]TJ/F17 7.9701 Tf 6.586 0 Td [(kt )]TJ/F17 7.9701 Tf 6.586 0 Td [(` p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k waystonishtherealization.Wecompletetheproofbysummingover k Lemma3.27. If pt>m ,thenfor k> 0 wehave P N k = t +1 k m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b m=t c)]TJ/F10 6.9738 Tf 9.88 0 Td [(k X j =0 kt jt + j + kt jt + j + kt j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(j )]TJ/F11 9.9626 Tf 9.963 0 Td [(jt p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(j : Proof. Again,werstassumethatforthelast k timestheurnisnonemptyandneutral,theevent A t follows.WethenusethesamemappingthatprovedLemma3.10.Whiletheresultingrealization isstillfromthe m;p )]TJ/F11 9.9626 Tf 10.053 0 Td [(k ;1 ;t urn,thistimetherealizationmusttaketheweight pt )]TJ/F11 9.9626 Tf 10.053 0 Td [(m ,whichis nolongerbetweentheinitialweight p )]TJ/F11 9.9626 Tf 9.751 0 Td [(k t )]TJ/F11 9.9626 Tf 9.751 0 Td [(m andzero.Thetotalnumberofrealizationstaking theweight pt )]TJ/F11 9.9626 Tf 10.115 0 Td [(m equals Q m;p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k pt )]TJ/F11 9.9626 Tf 10.116 0 Td [(m ,byLemma3.26.Themappingcanbereversed,andwe addthefactor t +1 k similarlybyLemma3.7,sincetheevents A 0 ;:::;A t areequallylikely. Whilewecouldsimilarlyarriveanotherformof G m;p ;1 ;t for pt>m viatheresultofLemma 3.27,itismorecomplicatedthanthezerocountform,anddoesnotaddressthevalueofthecrossing number .Thesamecouldbesaidforaformulainvolving N + aswell. 46

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Lemma3.28. If pt>m ,thenfor k> 0 wehave P N + k = t k m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k X j = d p )]TJ/F10 6.9738 Tf 6.226 0 Td [(m=t e k p )]TJ/F11 9.9626 Tf 9.962 0 Td [(j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t +1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(j m )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt + jt + j j : Proof. Webreakuparealization with N + k againasinLemma3.27: = TP k M k Q k P k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 M k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P 1 M 1 Q 1 : Ourmapthistimewillbeslightlydierent.Weshallmap tothe 0 fromthe m + k;p )]TJ/F11 9.9626 Tf 10.049 0 Td [(k ;1 ;t urn,with 0 = Q 1 M 0 1 Q k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 M 0 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q k M 0 k )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(TP k P k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P 1 R = Q 1 M 0 1 Q k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 M 0 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q k M 0 k P R 1 P R k T R : Since T R hasweight pt )]TJ/F11 9.9626 Tf 10.519 0 Td [(m 0 isarealizationforwhichtheweight pt )]TJ/F11 9.9626 Tf 10.519 0 Td [(m> 0istaken.This mappingissimilarlyinvertible.Wendthesegments Q i M 0 i asbefore,andwecanndthesegments P R i similarly.Thistime,thesmallestprexof 0 withweight )]TJ/F11 9.9626 Tf 7.749 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(it is Q 1 M 0 1 Q k M 0 k P R 1 P R i : Havingidentiedeachsegment,wecanundothemapandrecover from 0 .Thiscanbedonefrom anyrealizationfromthe m + k;p )]TJ/F11 9.9626 Tf 9.586 0 Td [(k ;1 ;t urnthattakestheweight pt )]TJ/F11 9.9626 Tf 9.586 0 Td [(m ,thereforethemapping isaone-to-onecorrespondence.Therefore,thereare Q m + k;p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k pt )]TJ/F11 9.9626 Tf 9.054 0 Td [(m suchrealizations.Theresult nowfollowsfromLemma3.26,afterwhichwereindexwithrespectto p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(j Remark .Ifwesettheindeterminateform0 = 0toequalone,thentheresultsofTheorems3.27and 3.28alsoholdfor k =0. 3.4. DistributionoftheGainforthe m;p ;1 ;t Urns. Fortheoriginal m;p ;1 ; 1urns,wenot onlyhavetheexpectedgain,butwealsohavethedistributionforthegainviaLemma3.4.Wecan alsondthedistributionofthegainforthe m;p ;1 ;t urnsbecausethereisalinkbetweengainand maximumweight,andforthe m;p ;1 ;t urnscountingthenumberofrealizationswithacertain maximumweightisnotadiculttask. Theorem3.29. Let m p ,and ` 0 begiven.Then P playergains ` usinganoptimalstrategy = m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Q m;p ` )]TJ/F11 9.9626 Tf 9.962 0 Td [(Q m;p ` +1 : 47

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Theorem3.29impliesthattheprobabilitytheplayergainsatleast ` equals )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q m;p ` ,and inparticulartheprobabilitytheplayergainsatleastmax f 0 ;pt )]TJ/F11 9.9626 Tf 10.37 0 Td [(m g equalsone.Themethodof proofistoshowabijectionbetweentherealizationsgainingexactly ` withtherealizationswitha maximumweightequalto ` .SincethisresultholdsundermoregeneralcircumstancesTheorem 4.18,wewillholdoontheproofuntilthen.Ouraimforthemomentisthefollowingresult,the distributionform Corollary3.30. Distributionform Forany m and p ,wehave G m;p ;1 ;t = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 pt X ` =1 Q m;p ` : ObservethatCorollary3.30givesanexactformfor G m;p ;1 ;t forany m and p ,includingan accuraterepresentationforthecase pt )]TJ/F11 9.9626 Tf 10.113 0 Td [(m> 0.When pt )]TJ/F11 9.9626 Tf 10.113 0 Td [(m> 0,wehave Q m;p ` = )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p for 1 ` pt )]TJ/F11 9.9626 Tf 9.495 0 Td [(m ,andbecauseofthiswecanhidethegainassociatedwiththeinitialvalueoftheurn insidethesum.Inparticular,when t =1wemayrewritetheresultofTheorem3.2inawaythat matchestheformofCorollary3.30: G m;p = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 m + p min f k;m g : Thedistributionsofthegainsoftheurnswith t =2and m + p =5ballsaregiveninTable3.1. Table3.1. Distributionofthegainforthe m;p ;1 ; 2urns,with m + p =5. urn012345678910 ; 0;1 ; 210000000000 ; 1;1 ; 23 = 51 = 51 = 500000000 ; 2;1 ; 203 = 102 = 51 = 51 = 10000000 ; 3;1 ; 200003 = 53 = 101 = 100000 ; 4;1 ; 200000004 = 51 = 500 ; 5;1 ; 200000000001 48

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4. TheValueofthe m;p ; s;t Urn Wenowtackletheproblemofndingthevalueofthe m;p ; s;t urn,initiallywith t=s apositive rational.TheCrossingLemmaisnolongeratourdisposal,sowelosetheresultofCorollary3.8 withit.Thus,azerocount,negativebinomial,orbinomialformappearstobeoutofourreach. Asforthedistributionform,Theorem3.29doesholdunderthesegeneralcircumstancesTheorem 4.18,buttheobjectsinvolvedareverydicultorimpossibletoobtaindirectly.Thus,weneed anotherapproach.Tondthatapproach,wetakeanotherlookatthe m;p ;1 ;t urns.Indeed, aratherprimitiveapproachgivesanotherformfor G m;p ;1 ;t ,the crossingsform Lemmas4.5, 4.6.Usingthesameapproach,wecancalculate G m;p ; s;t Theorems4.10,4.11exactlyforany s and t ,butnotwithoutsomechallengesalongtheway.Forsomespecialcases,wecansimplify thecrossingsform.Inparticular,when pt = ms wecanwrite G m;p ; s;t inaformTheorem4.15 resemblingthezerocountformof G m;p ;1 ;t When r = t=s isirrational,thecrossingsapproachalsoworksperfectly.Thedicultyliesin actuallywritingaformulafortheexpectedgain.Usingarationalapproximationandthecontinuity of G m;p ; s;t in t ,wecanwrite G m;p ;1 ;r inaformnotinvolvingalimitTheorem4.17when r isirrational. Finally,weshowthattheprobabilitythat k isgainedequalstheprobabilitythatthemaximum weightoftheurnequals k .Therefore,theresultsfortheexpectedgainusinganoptimalstrategy alsoapplytotheexpectedmaximumweightorminimumweight,bytheReversalLemmatheurn achievesduringplay. 4.1. AMorePrimitiveFormulafor G m;p ;1 ;t Thezerocount,negativebinomial,andbinomialformsof G m;p ;1 ;t with t apositiveintegeralldependupontheCrossingLemma,in particularCorollary3.8,whichreducedcalculatingtheexpectedgaintocountingthenumberof timestheurncouldbeneutral.SincetheresultoftheCrossingLemmadoesnotholdforthe m;p ; s;t urns,andthedistributionformwillbeexceedinglycomplicated,weneedtondanapproachthatcalculates G m;p ;1 ;t withoutthehelpoftheCrossingLemma.Todothat,weneed togureouttheeventoreventsresultingingainsfortheplayer. 49

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4.1.1. Crossings. Inourrstderivationof G m;p ;1 ;t ,wenotedthatiftheurnweightwaszero,the playergainedanaverageof t= 2untilthenexttimetheurnwasneutral.Now,wewillindicatethe preciseeventsthatexplicitlygivethevalueforthe m;p ;1 ;t urn.A permanentgain thatis,these gainscombinedwillequalthetotalgainfortherealizationcanbemadeinoneofthreeways:First, if pt )]TJ/F11 9.9626 Tf 9.818 0 Td [(m> 0,thensince X m + p =0itisacertaintythattheplayerwillgainatleast pt )]TJ/F11 9.9626 Tf 9.818 0 Td [(m .The remainingpermanentgainsaretheresultoftwosimilarevents,thatweshallcalldown-crossings andup-crossings. Denition4.1. Suppose X n = t )]TJ/F11 9.9626 Tf 10.767 0 Td [(j and X n +1 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(j .Ifthezero-passstrategyisused,then thiseventisa down-crossing to )]TJ/F11 9.9626 Tf 7.749 0 Td [(j if0 j t )]TJ/F8 9.9626 Tf 10.451 0 Td [(1.Withthezero-betstrategy,thiseventisa down-crossingto )]TJ/F11 9.9626 Tf 7.749 0 Td [(j if1 j t Whenthezero-passstrategyisused,eachdown-crossingto )]TJ/F11 9.9626 Tf 7.749 0 Td [(j willresultintheapermanentgain of j fortheplayer.Thisisbecausetheplayerwillwaituntiltheurnweightisatleastzerobefore placinganotherbet. Thenotionofanup-crossingisdenedsimilarly. Denition4.2. Suppose X n = i )]TJ/F11 9.9626 Tf 10.349 0 Td [(s ,and X n +1 = i .Ifthezero-passstrategyisused,thenthis eventisan up-crossing to i if1 i s .Withthezero-betstrategy,thiseventisanup-crossingto i if0 i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1. Forthe m;p ;1 ;t urnswiththezero-passstrategy,wehaveanup-crossingwhenever X n =0and thenextballdrawnfromtheurnisa )]TJ/F8 9.9626 Tf 7.749 0 Td [(1."Sincetheplayerwillnotbetonthisball,andwillbet untiltheweightoftheurnisnonpositive,theplayerwillthereforegainoneasaresultofthisevent. Anyremaininggainovertheseriesofbetswillbecountedbythefollowingdown-crossing.Since zeroisunfavorablewiththezero-passstrategy,thelastcrossingwillbealwaysbeadown-crossing. Usingtheseup-anddown-crossings,wecalculateyetanotherformulafor G m;p ;1 ;t ,the crossingsform 4.1.2. TheCrossingsFormof G m;p ;1 ;t Usingthezero-passstrategy,if pt )]TJ/F11 9.9626 Tf 10.296 0 Td [(m 0and p> 0 implying m t ,thenthetotalnumberofup-crossingsoverallrealizationsis t t +1 p X k =1 kt + k k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : 50

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Figure4.1. Forthisrealizationfromthe ; 4;1 ; 2urn,thethird,sixth,and twelfthballsblackresultinup-crossingswhenthezero-passstrategyisused,with eachresultinganagainofonefortheplayer.Thefourthandthirteenthballsare down-crossingsto )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,witheachresultinginagainofoneshowninblackforthe player.Theeighthballisadown-crossingtozero,butdoesnotresultinagainfor theplayer.Thisrealizationgainsvefortheplayerwhenthezero-passstrategyis used. Weassociateapermanentgainofonewitheachoccurrenceofanup-crossing. Foreach0 i t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1,thetotalnumberofdown-crossingsto )]TJ/F11 9.9626 Tf 7.749 0 Td [(i is p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 kt + k + i k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Weassociatethepermanentgainof i withthese.Theup-anddown-crossingsofarealizationfrom the ; 4;1 ; 2urnareshowninFigure4.1. Lemma4.3. If pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m 0 ,then m + p p G m;p ;1 ;t = t t +1 p X k =1 kt + k k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k + t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 i kt + k + i k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Ifwehave pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m> 0,thenthetotalnumberofup-crossingsis t t +1 b m=t c X k =1 kt + k k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : Fordown-crossings,acomplicationdevelops.Afewdown-crossingsmayappearbeforetheurnis rstneutral.However,whichandhowmanyareeasilydetermined.Theveryrstpossibledowncrossingcanbefoundbydrawingnothingbut+ t "ballsuntiltheurnisnonpositive.Thus,ifthat rstonecrossesdownto )]TJ/F11 9.9626 Tf 7.749 0 Td [(K ,then pt )]TJ/F11 9.9626 Tf 9.63 0 Td [(m thus )]TJ/F11 9.9626 Tf 7.748 0 Td [(m and )]TJ/F11 9.9626 Tf 7.749 0 Td [(K arecongruentmodulo t .Thenthere 51

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areextradown-crossingsto )]TJ/F11 9.9626 Tf 7.749 0 Td [(K )]TJ/F11 9.9626 Tf 7.748 0 Td [(K +1 ;:::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,and0.Thusifwedene K astheresidueof m modulo t anddene,for0 i t )]TJ/F8 9.9626 Tf 9.962 0 Td [(1, .1 M i = 8 > > < > > : b m=t c ; if0 i K b m=t c)]TJ/F8 9.9626 Tf 16.604 0 Td [(1 ; otherwise, wecansaythetotalnumberofdown-crossingsto )]TJ/F11 9.9626 Tf 7.748 0 Td [(i equals M i X k =0 kt + k + i k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Likebefore,weassociateapermanentgainof i witheachdown-crossingto )]TJ/F11 9.9626 Tf 7.749 0 Td [(i .Sinceeachrealization isequallylikely,wehavewhatweneedtocalculate G m;p ;1 ;t Lemma4.4. Suppose pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m 0 .With K and M i asdenedin .1 ,wehave m + p p G m;p ;1 ;t = m + p p pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + t t +1 b m=t c X k =1 kt + k k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k + t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 M i X k =0 i kt + k + i k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Notethatwehaveincludedthecase pt = m aswell.Thedierenceintheargumentisthatthere isanonexistentdown-crossingtozeroaddedsince K =0,coming before therstdrawfromthe urn.However,itisassociatedwiththevanishing i =0term. Ifourplayerdecidestotakethezero-betapproach,theneutralurnsarenowdeemedfavorable. Theup-crossingsnowrunfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(1to0,butsincetheurnisnotguaranteedtoreturntobackto theweight )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,wedonotassociateapositivegainwiththeseevents.Therefore,wecanignorethem altogether.Adown-crossingto )]TJ/F11 9.9626 Tf 7.749 0 Td [(i ,thistimewith1 i t ,stillresultsinapermanentgainof i fortheplayer.Wenowconstructamoreocial"setofformulasthatturnouttoberedundantto theformulasofLemmas4.3and4.4,thoughthenexttworesultsareinabetterformsincewecan ignoretheup-crossings. Lemma4.5. CrossingsFormfor m pt Suppose pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m 0 .Thenwehave G m;p ;1 ;t = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t X i =1 p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 i kt + k + i k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : 52

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Lemma4.6. CrossingsFormfor m pt Suppose pt )]TJ/F11 9.9626 Tf 10.248 0 Td [(m 0 .With K and M i asdenedin .1 ,wehave G m;p ;1 ;t = pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t X i =1 M i X k =0 i kt + k + i k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Lemmas4.4and4.6areidenticalresults,since K 0,unlikethezerocountestimate. 4.2. TheExpectedGainforthe m;p ; s;t Urnwith t=s Rational. Withthismoreprimitive approach,wecanobtainformulasforthevalueofthe m;p ; s;t urns,foranyarbitrarypositive rational s and t .Withoutlossofgenerality,wewillimposethefollowingrestrictionson s and t :We requirethat s and t arepositiveintegerswithgcd s;t =1,andwealsorequirethat ms )]TJ/F11 9.9626 Tf 10.118 0 Td [(pt 0. Wecanexpandbacktothecaseswith t=s rationalviaLemma2.1,whilewecanextendtothecase ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt< 0viatheAntiurnTheorem. Twocomplicationsoccurasaresultofenlarging s .First,wenowwillhavemultipleup-crossings todealwith.Anothercomplexityarisesinthecycle"-theurnweightincreasesby s modulo t + s eachtimeaballisdrawn,andtheorderingonlyrelateswellwiththeregularintegerorderwhen s or t equalsone.Formulaswillbemorecomplex,andbemorediculttosimplify,asaresult.So, ourrsttaskshallbetodeneanorderon f)]TJ/F11 9.9626 Tf 12.73 0 Td [(t; )]TJ/F11 9.9626 Tf 7.749 0 Td [(t +1 ;:::;s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;s g tosimplifynotation. Denition4.7. The crossingorder R on f)]TJ/F11 9.9626 Tf 12.73 0 Td [(t; )]TJ/F11 9.9626 Tf 7.748 0 Td [(t +1 ;:::;s g isdenedasfollows:Foreach i 2 f)]TJ/F11 9.9626 Tf 12.73 0 Td [(t;:::;s g ,thereexistsaunique c i 2f 0 ;:::;s + t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g suchthat i c i t mod s + t .Then .2 iRj ifandonlyif c i c j : Notethat c )]TJ/F10 6.9738 Tf 6.227 0 Td [(t = c s = t + s )]TJ/F8 9.9626 Tf 10.485 0 Td [(1.Thisambiguityshallnotbeaproblem,as )]TJ/F11 9.9626 Tf 7.748 0 Td [(t forthezero-bet strategyand s forzero-passwillnotbeusedsimultaneously. Themainreasonfortheintroductionofthecrossingorder R isthattherstpossibleup-crossing maycomemid-cyclethatis,beforethersttimetheurncouldbeneutral.Ifthisrstup-crossing 53

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isto K ,thenwewillhaveanextracrossingforeach i satisfying iRK .Therstpossibleup-crossing occurswhenonly )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsaredrawnuntilafavorableurnisreached.Therefore, K and pt that is, pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms willbecongruentmodulo s Next,wedeterminewhentheurncanpossiblybeanywhereintherange f)]TJ/F11 9.9626 Tf 12.731 0 Td [(t;:::;s g Denition4.8. Foreachinteger )]TJ/F11 9.9626 Tf 7.749 0 Td [(t i s ,theuniquenon-negativeintegers a i and b i satisfying .3 a i + b i
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Proof. Recallthat b i t )]TJ/F11 9.9626 Tf 8.738 0 Td [(a i s = i .If b i = b j ,wehave i )]TJ/F11 9.9626 Tf 8.738 0 Td [(j = a j )]TJ/F11 9.9626 Tf 8.738 0 Td [(a i s ,whichmeans i = j ,sinceboth i and j arebetween1and s )]TJ/F8 9.9626 Tf 8.628 0 Td [(1.Sincetheurniscrossingupfromanegativevalue,wehave b i > 0.We alsohave b i 0,thenthereisagainof pt )]TJ/F11 9.9626 Tf 8.989 0 Td [(ms obtainedthroughtherstseriesofbets.Otherwise,gainsaremadeviaup-anddown-crossings. Foranup-crossingto i ,theremustbe kt + a i )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "ballsand ks + b i + t "ballsleftintheurnat itscompletion,forsomevalid k .Theup-crossingconsistsofone )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "ball,thustherehavebeen m )]TJ/F11 9.9626 Tf 10.215 0 Td [(kt )]TJ/F11 9.9626 Tf 10.215 0 Td [(a i )]TJ/F8 9.9626 Tf 10.215 0 Td [(1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsand p )]TJ/F11 9.9626 Tf 10.216 0 Td [(ks )]TJ/F11 9.9626 Tf 10.215 0 Td [(b i + t "ballsdrawnfromtheurnbeforethecrossing.A similarargumentappliesforthedown-crossings,theonlydierencebeingthatthedown-crossing consistsofalone+ t "ballinstead.Wecannowcalculate G m;p ; s;t Theorem4.10. Suppose pt )]TJ/F11 9.9626 Tf 9.949 0 Td [(ms 0 .Let K betheresidueof pt modulo s .For )]TJ/F11 9.9626 Tf 7.749 0 Td [(t i s dene .4 M i = 8 > > < > > : b p=s c ; if iRK b p=s c)]TJ/F8 9.9626 Tf 16.605 0 Td [(1 ; otherwise, wherethecrossingorder R isdenedby .2 .Then m + p p G m;p ; s;t = s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 M i X k =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i + t X j =1 M )]TJ/F10 6.9738 Tf 6.227 0 Td [(j X k =0 j k t + s + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : TheresultforTheorem4.10isviathezero-betstrategy.Thus,weshowthe i =0term,andhide the j =0term. Example .Takingtheneutral ; 4;4 ; 3urn,wehave K =0,and M i =0forall i .Thenusingthe entriesfromTable4.1weobtain 35 G ; 4;4 ; 3= 5 3 1 1 +2 3 2 3 2 +3 1 1 5 3 # + 2 1 4 2 +2 4 2 2 1 +3 6 3 0 0 # =154 : Thus G ; 4;4 ; 3=154 = 35=4 : 4. Usingthezero-passstrategyweobtainanotherform. 55

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Theorem4.11. Suppose pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms 0 .With K and M i asdenedin .4 ,wehave m + p p G m;p ; s;t = s X i =1 M i X k =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i + t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 M )]TJ/F10 6.9738 Tf 6.227 0 Td [(j X k =0 j k t + s + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Forthe ; 4;4 ; 3urn,the3 )]TJ/F15 7.9701 Tf 5.48 -4.379 Td [(6 3 )]TJ/F15 7.9701 Tf 10.959 -4.379 Td [(0 0 termwith j = t isreplacedbythe i = s termof4 )]TJ/F15 7.9701 Tf 5.48 -4.379 Td [(6 4 )]TJ/F15 7.9701 Tf 10.959 -4.379 Td [(0 0 Bothtermscontribute60tothesum,sowestillget G ; 4;4 ; 3=4 : 4. Weclosebygivingexplicitformulasforsomespecialcases,beginningwiththeoneball"formulas. Theorem4.12. Let N 1 =max f n 2 Z : ns t g ,and N 2 =max f n 2 Z : nt s g .Thenfor m> 0 G m; 1; s;t = N 1 +1 m +1 t )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(sN 1 2 ; andfor p> 0 G ;p ; s;t = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(s + N 2 +1 p +1 s )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(tN 2 2 : Proof. Weshallusethezero-betstrategytocalculate G m; 1; s;t .Theurnwillbenonnegative onlywhenthe+ t "ballisoneofthelast N 1 +1ballsdrawnfromtheurn.Workingwiththese realizations,theplayerwillstartbettingoncethiseventoccurs,andwilldraw i )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls,where 0 i N 1 ,thenthe+ t "ball,afterwhichtheplayerwillstopbetting.Thustheplayer'sexpected gainis m +1 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 N 1 X i =0 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(is = N 1 +1 m +1 t )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(sN 1 2 : ThesecondresultfollowsfromthisbytheAntiurnTheorem. Ournalformulaisforthecasewherethereisquiteadisparitybetween s and t Theorem4.13. Supposethat s pt ,and m> 0 .Then G m;p ; s;t = t m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p X k =0 k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Proof. If s pt ,thentherearenocontributingdown-crossingswiththezero-passstrategy.The onlyup-crossingsoccurwiththedrawofthelast )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ball.Ifthereare k + t "ballsleftafter thelast )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballhasbeendrawn,thentheplayerwillgain kt .Foreach k ,thereare )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p )]TJ/F17 7.9701 Tf 6.586 0 Td [(k )]TJ/F15 7.9701 Tf 6.586 0 Td [(1 p )]TJ/F15 7.9701 Tf 6.587 0 Td [(1 56

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realizationsresultinginagainof kt .Thuswesumupanddividebythenumberofrealizations, )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p G m;p ; s;t isnonincreasingas s increases,andtheformulagivenbyTheorem4.13doesnot dependon s outsideoftheassumption s pt .Thus,if pt> 0wehaveinf s G m;p ; s;t > 0,with any s pt givingtheinnum.Thisisbecausetheplayerwillsimplywaituntilallofthe )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls aredrawn,andcollectany+ t "ballsthatmaybeleft. TheAntiurnTheoremprovidesuswithaformulaattheoppositeendofthespectrum. Corollary4.14. Supposethat t ms ,and p> 0 .Then G m;p ; s;t = pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms + s m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m X k =0 k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Theexpectedgainsforsomeoftheurnswith s =4and t =3aregiveninTable4.2. Table4.2. Expectedgainforsome ; ;4 ; 3urns,roundedtotwodecimalplaces. Gainsforneutralurnsareinbold. p 012345678 G ;p ;4 ; 3 0 3691215182124 G ;p ;4 ; 301.53.676.25911.8314.7117.6320.56 G ;p ;4 ; 3012.334.16.48.9511.6414.4217.24 G ;p ;4 ; 300.751.72.9 4.4 6.488.8711.4314.11 G ;p ;4 ; 300.61.332.233.314.726.588.811.24 G ;p ;4 ; 300.51.101.802.643.6856.678.73 G ;p ;4 ; 300.430.931.512.193.004.005.23 6.73 G ;p ;4 ; 300.380.811.31.872.533.324.285.42 4.3. TheSpecialCase pt = ms Asimplerformulaforthevalueofthegeneralurnseemstobe diculttoobtain.However,fortheneutralurnswith pt = ms ,weareabletodosomesimplication. Theorem4.15. Suppose pt = ms .Then G m;p ; s;t = m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t + s 2 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i : Proof. Weusethezero-betformulagivenbyTheorem4.10.Webeginbyreversingtherolesthe twobinomialcoecientsinthedoublesumrepresent,byliterallyreversingtherealization.Inthis manner,theballsleftintheurnbecometheballsdrawnfromit,andviceversa.Uponthisreversal, 57

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eachup-crossingto i becomesanup-crossingto s )]TJ/F11 9.9626 Tf 10.328 0 Td [(i instead.Whenweputthegainsassociated withthetwoup-crossingstogether,weendupwithagain, s ,thatnolongerdependson i .Interms ofthealgebra,wewillshowthat s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =1 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i .5 = s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =1 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i : Startingwiththelefthandsideof.5,reindexingtheoutersumwithrespectto s )]TJ/F11 9.9626 Tf 10.006 0 Td [(i givesus thesum s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =1 p=s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i k t + s + c s )]TJ/F10 6.9738 Tf 6.226 0 Td [(i ks + b s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i : Next,wereindextheinnersumwithrespectto p=s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k .Thisgives s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =1 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i k t + s + t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ks + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(b s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F8 9.9626 Tf 9.963 0 Td [( t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F8 9.9626 Tf 9.962 0 Td [( s )]TJ/F11 9.9626 Tf 9.962 0 Td [(b s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i ; since ms = pt .Bythedenitionofthecrossingconstants, b s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i t )]TJ/F11 9.9626 Tf 9.838 0 Td [(a s )]TJ/F10 6.9738 Tf 6.226 0 Td [(i s = s )]TJ/F11 9.9626 Tf 9.839 0 Td [(i .Wethereforehave that s )]TJ/F11 9.9626 Tf 9.963 0 Td [(b s )]TJ/F10 6.9738 Tf 6.226 0 Td [(i t )]TJ/F8 9.9626 Tf 9.963 0 Td [( t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i s = i: Wehave0
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whileasimilarmanipulationgives t p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(t ks + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(t m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(t )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .7 = t p=s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 k t + s ks m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = s p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s ks m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks : Together,.5,.6,and.7implythat 2 m + p p G m;p ; s;t = s s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i .8 + t t X j =1 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Wecancombinethetwosumsrathereasilyuponnoticingthat s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 p=s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i .9 = t X j =1 p=s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Theformersumisthetotalnumberofzero-betup-crossings,andthelattersumisthetotal numberofdown-crossings.Sincezeroisfavorablewiththezero-betstrategy,therstcrossingif thereisoneisadown-crossing.Thedown-andup-crossingsalternatethroughout,butthelast crossingmustbeanup-crossing,again,sincezeroisfavorable.Thus,weconcludethat.9holds. Combiningthetwosums,wethenhave 2 m + p p G m;p ; s;t = t + s s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i ; andwecompletetheproofbydividingthroughby2 )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p Fortheneutral ; 4;4 ; 3urn,wehaveanewcalculation,butthesameresult: G ; 4;4 ; 3= 1 35 7 2 0 0 6 4 + 5 3 1 1 + 3 2 3 2 + 1 1 5 3 # =4 : 4 : 59

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Corollary4.16. Suppose pt = ms .Then G m;p ; s;t = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t + s 2 t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Proof. From.8,weuse.9theotherwaytocombinethetwodoublesums.Then,usingthe intermediatestepof.7wecanmovethe j = t termtothe j =0term.Thus, 2 m + p p G m;p ; s;t = t + s t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 p=s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ; andweagaindividethroughby2 )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p Thisgivespredictably G ; 4;4 ; 3= 1 35 7 2 0 0 6 3 + 2 1 4 2 + 4 2 2 1 # =4 : 4 : Remark .Acombinatorialproofofthisspecialcaseisasfollows:Forarealization ,weexaminethe gainsmadefrom usingthezero-passstrategy,andthegainsmadefromthereversedrealization R = ;m + p ]usingthezero-betstrategy.Fromtheperspectiveof ,therstcrossingisan up-crossing.Eachup-crossinggivesapermanentgainof i from and s )]TJ/F11 9.9626 Tf 10.648 0 Td [(i fromitsassociated up-crossingin R ,for1 i s .Thefollowingdown-crossinggivesapermanentgainof j from and t )]TJ/F11 9.9626 Tf 10.7 0 Td [(j from R for0 j t )]TJ/F8 9.9626 Tf 10.7 0 Td [(1.Sincewehavethepairingofup-anddown-crossings, wecanassociatethegainsfromthedown-crossingwiththeprecedingup-crossing.Thus,foreach up-crossingwecanassociateacombinedpermanentgainof t + s from and R .Summingupover all ,wedivideby2,sinceeachrealizationiscountedtwice. 4.4. UsingRationalApproximationtoCalculate G m;p ; s;t When t=s IsIrrational. The approachthatyielded G m;p ; s;t when t=s isrationalwillalsoworkwhen t=s isirrational.The dicultyarisesinwritingthevalueinareadableform.Hence,weneedtondadierentwayto express G m;p ; s;t .Alogicalstrategywouldbetousesomesortofrationalapproximation,since wehavethecontinuityof G m;p ; s;t inboth s and t .Forastart,wecouldsaythatif f r n g 1 n =1 is asequenceofrationalnumbersconvergingto r = t=s ,then G m;p ; s;t = s G m;p ;1 ;r = s lim n !1 G m;p ;1 ;r n ; 60

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butgiventhecomplicatednatureoftheformulasfor G m;p ;1 ;r n ,thisisnotasatisfactoryrepresentationof G m;p ; s;t .Instead,wetruncatethesequence r 1 ;r 2 ;::: atanappropriatespot N andusetheapparatusassociatedwiththe m;p ;1 ;r N urntocalculateandexpress G m;p ;1 ;r Supposewehaveanurnwith m )]TJ/F8 9.9626 Tf 7.748 0 Td [(1"ballsand p + r "balls,thistimewith r positiveand irrational.Weshallusearationalapproximation y=x to r with y=x>r thoughwecouldjustas welluse y=xr ,thenwearehalfwaythere: k y=x )]TJ/F11 9.9626 Tf 9.567 0 Td [(j>kr )]TJ/F11 9.9626 Tf 9.567 0 Td [(j 0,soourconcernis choosing y=x>r sothat kr )]TJ/F11 9.9626 Tf 8.724 0 Td [(j j )]TJ/F11 9.9626 Tf 8.724 0 Td [(kr )]TJ/F1 9.9626 Tf 8.725 8.069 Td [()]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(j )]TJ/F11 9.9626 Tf 8.725 0 Td [(k y=x = k y=x )]TJ/F11 9.9626 Tf 10.389 0 Td [(r .Ifwereplace,ontheright-handside, k with p ,wewillstillhavetheaboveequation satised.Setting =min f j )]TJ/F11 9.9626 Tf 9.963 0 Td [(kr :0 j m; 0 k p;j )]TJ/F11 9.9626 Tf 9.963 0 Td [(kr> 0 g andchoosing y=x sothatthefollowingconditionsaresatised: 61

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A1 y x >r A2 x + y>m + p A3 y x )]TJ/F11 9.9626 Tf 9.962 0 Td [(r< p wewillhave.10satised.Withthecorrecttypesofcrossingsintherightspots,wecanusethese crossingswiththeirrationalurn. Nowwedealwiththegainsassociatedwiththesecrossings.Usingthe m;p ; x;y urn'scrossings, ifthereisanup-crossingto i ,thenthereare c i ballsleft, b i ofwhichare+ y "and a i are )]TJ/F11 9.9626 Tf 7.749 0 Td [(x ,"and b i y )]TJ/F11 9.9626 Tf 8.618 0 Td [(a i x = i .Dividingthroughby x ,forthe m;p ;1 ;y=x urn,anup-crossingto i=x gives b i + y=x ballsand a i )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balls.Thisisindirectcorrespondencewithanup-crossinginthe m;p ;1 ;r urn with b i + r "ballsand a i )]TJ/F8 9.9626 Tf 7.748 0 Td [(1"ballslefttodraw.Thatis,thepermanentgainassociatedwiththe m;p ;1 ;r urnis b i r )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i = b i r )]TJ/F11 9.9626 Tf 11.384 6.74 Td [(y x + b i y x )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i = b i r )]TJ/F11 9.9626 Tf 11.384 6.74 Td [(y x + i x : Similarly,withadown-crossingto )]TJ/F11 9.9626 Tf 7.749 0 Td [(j inthe m;p ; x;y urn,wehavethepermanentgainof a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j r = a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j y x + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j y x )]TJ/F11 9.9626 Tf 9.963 0 Td [(r = j x + b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j y x )]TJ/F11 9.9626 Tf 9.963 0 Td [(r associatedwiththe m;p ;1 ;r urn.Therefore,wehavethefollowing: Theorem4.17. Suppose r> 0 isirrational, pr )]TJ/F11 9.9626 Tf 10.377 0 Td [(m< 0 ,andthat x and y arepositiveintegers satisfying A1-A3 .Dene a i b i and c i sothat a i y )]TJ/F11 9.9626 Tf 9.027 0 Td [(b i x = i a i + b i = c i 0,viatheAntiurnTheorem, byobservingthat G p;m ; r; 1= rG p;m ;1 ; 1 =r .Weomitthedetails. 4.5. TheRelationshipBetweenGainandMaximumWeight. Weusethenow-expanded notation Q s;t m;p ` todenotethenumberofrealizationsfromthe m;p ; s;t urnforwhich X n ` forsome n )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(InthecontextofearlierresultslikeTheorem3.26, Q m;p ` = Q ;t m;p ` 62

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Theorem4.18. Foranynonnegative s t ,andintegers m p ` 0 ,usinganyoptimalstrategywe have P playergainsatleast ` = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q s;t m;p ` : Proof. Weshowthatthenumberofrealizationsforwhichtheplayergains ` equals Q s;t m;p ` )]TJ/F11 9.9626 Tf -424.251 -19.929 Td [(Q s;t m;p ` +1,thenumberofrealizationsforwhichmax n X n = ` .Fromthistheresultimmediately follows. Weneedonlytoshowtheresultforthetwomainstrategies.Supposethatthezero-betstrategy isused.Wedivideeachrealization intotwoparts, + and )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(.Thepart )]TJ/F8 9.9626 Tf 10.215 -3.615 Td [(shallconsistofthe ballsaccepted,and + shallconsistoftheballsnotaccepted.Inboth,thepositionsoftheballs relativetoeachotherarepreserved.Wethenformtherealization^ = + )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [( R .Thebreakdown ofarealizationfromthe ; 5;2 ; 3urnisdepictedinFigure4.2. Figure4.2. Forthisrealization fromthe ; 5;2 ; 3urn,thedashedportionsform + ,andthesolidportionsform )]TJ/F8 9.9626 Tf 10.225 -3.615 Td [(whenthezero-betstrategyisused.Thisrealizationgainssixusingthezero-betstrategy.Themappedrealization^ = + )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [( R notpicturedreachesamaximumweightofsix,aftersixballshavebeendrawn. Suppose resultsinagainof ` withthezero-betstrategy.Weclaimmax n X n ^ = ` .Clearly, ^ takesaweightatleast ` ,andactuallytakestheweight ` ,since )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [( R containsalloftheaccepted balls,hasweight ` ,andendstherealization^ .Twosimpleobservationsshowthatthemaximum weightis ` .Callamaximalstringofconsecutivebetsa bettingsession ,andsimilarlydenea passing session .Thepart )]TJ/F8 9.9626 Tf 10.276 -3.615 Td [(isthereforetheconcatenationofallofthebettingsessionsof ,while + is 63

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theconcatenationofthepassingsessions.Foreachbettingsession,theurnweightisnonnegative atthebeginningin ,andtheurnweightisnegativeattheend,exceptforpossiblythelast session,forwhichtheurnweightisnonnegativeattheend.Viewing )]TJ/F8 9.9626 Tf 11.04 -3.616 Td [(asitsownrealization, min n X n )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(=0,withthisminimumachievedattheendof )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(.Reversing,max n X n )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [( R = ` withthisweightattainedat n =0.Similarly,eachpassingsessionbeginswithanegativeweightand endswithanonnegativeweight.Therefore,asitsownrealization,max n X n + = ` )]TJ/F8 9.9626 Tf 8.15 0 Td [( pt )]TJ/F11 9.9626 Tf 8.15 0 Td [(ms ,with thismaximumweightachieved only attheend.Inthecontextof^ ,theurnweightattheconclusion of + is ` ,andfurthermorethisisthersttimetheweight ` istaken.Therefore,max n X n ^ = ` Tocompletetheproof,weshowthatthemappingisaone-to-onecorrespondence.Suppose^ satisesmax n X n ^ = ` .Wendthersttimetheweight ` istaken,andsplit^ intotwoparts. Theballsprecedingthispointweshallcall + .Theballsfollowingthispointarerstreversed,and thenwecallit )]TJ/F8 9.9626 Tf 6.725 -3.616 Td [(.Wethenconstructtherealization bythefollowingdeterministicconstruction: Startingwith X 0 = pt )]TJ/F11 9.9626 Tf 10.577 0 Td [(ms ,wetakethe n th ballfrom + withoutreplacementif X n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 < 0, otherwisewetakethe n th ballfrom )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(,untilallballshavebeenused.Notethatthepropertiesof + and )]TJ/F8 9.9626 Tf 9.806 -3.616 Td [(preventascenarioinwhich,forexample, X n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 < 0and + isempty.Byconstruction, alloftheacceptedballsfrom comefrom )]TJ/F8 9.9626 Tf 6.724 -3.615 Td [(,whilealloftheballsnotacceptedcomefrom + Therefore, willgain ` viathezero-betstrategy. Notethatthesegments )]TJ/F8 9.9626 Tf 11.518 -3.616 Td [(and + switchroles,andthestrategyswitchestothezero-pass strategy,whentheantiurnmapisapplied.Thus,theresultunderthezero-passstrategynowfollows fromthezero-betresultandtheAntiurnTheorem.Ifwewantamoredirectway,wecanusethe sameconstructionasbefore.Create )]TJ/F8 9.9626 Tf 6.725 -3.616 Td [(, + ,and^ inthesamemanner,onlythistimeinassociation withthezero-passstrategy.Theonlymajordierenceistondthe last timetheweight ` istaken in^ beforesplitting.Weomittheremainingdetails. Corollary4.19. Thedistributionofthegainforthe m;p ; s;t urndoesnotdependonthechoice ofoptimalstrategy. Unfortunately, Q s;t m;p ` ,undergeneralcircumstances,appearsextremelydiculttocalculate. Q s;t m;p ` )]TJ/F11 9.9626 Tf 9.502 0 Td [(Q s;t m;p ` +1canbecalculatedusingthegeneralizedballotnumbersdiscussedinChapter 6,butaformulafortheexpressionwouldbeverycomplicated,andinadditionwillnotbeexplicit ingeneral. 64

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5. Asymptotics Wenowexaminewhatkindofgaincanbeexpectedforlargevaluesof m and p undervarious circumstances.Throughoutthissection,weshallassumethat s and t arenonnegativeintegerswith gcd s;t =1,and ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt = h t + s 0,unlessspecicallystatedotherwise. Recallthebinomialformfor G m;p ;1 ;t ,fromTheorem3.15: G m;p ;1 ;t = )]TJ/F11 9.9626 Tf 9.636 6.74 Td [(t 2 + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p X k =0 m + p +1 k t p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : .1 When s> 1,wehavethemoreunwieldycrossingsformgivenwithTheorem4.10.Ourmaintask willbetomold G m;p ; s;t intoaformmorelike.1,namelythesum m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p X k =0 m + p +1 k t s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ; .2 whichweshallcallthe pseudo-binomialform ,asitisnotanaccurateformfor G m;p ; s;t .Mostof thetime,itwillbecloseenough.Let b n;q;k = n k q k )]TJ/F11 9.9626 Tf 9.963 0 Td [(q n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k ;B n;q;m = m X k =0 b n;q;k : When0 q 1, b n;q;k and B n;q;m arerelatedtoabinomialdistribution.Here, p X k =0 m + p +1 k t s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k = t + s m + p +1 s p t m +1 B m + p +1 ; s t + s ;p : Inthisform,wemayusetoolsassociatedwithbinomialdistributions.ThisincludestheCentral LimitTheorem,whichwestatenow. Theorem5.1. CentralLimitTheorem Let f X n g beasequenceofindependentidenticallydistributedrandomvariableswith 0
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Asaconsequence,forabinomialdistribution B n;q with n trials,probabilityofsuccess q ,mean nq = ,andvariance nq )]TJ/F11 9.9626 Tf 9.962 0 Td [(q = 2 < 1 ,wehavefor n largeenoughthat .3 P B )]TJ/F11 9.9626 Tf 9.963 0 Td [( x 1 p 2 Z x e )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 2 = 2 dt: Weshalldenotethedistributionfunctionforthestandardnormalas x = 1 p 2 Z x e )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 2 = 2 dt: WeshallfrequentlyuseStirling'sFormulaaswell: z != z e z p 2 z )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ o =z : Inparticular,weuseittoanalyzethetotalnumberofrealizations, )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p Lemma5.2. Suppose ms )]TJ/F11 9.9626 Tf 9.662 0 Td [(pt = h t + s = o )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(max f m;p g ,andlet A bealargeconstant.Thenfor p largeenough, m + p p = t + s s p t + s t m r t + s 2 pt 1+ h p p 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m m )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A : Proof. m + p p = r m + p 2 mp m + p p p m + p m m )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A ; andobservethat m + p p = t + s s 1+ h p ; m + p m = t + s t 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m : 5.1. AnalyzingthePseudo-BinomialForm. Wewillnowanalyzethepseudo-binomialform m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 2 t + s m + p +1 s p t m B m + p +1 ; s t + s ;p : Westartwiththeassumptionthat ms )]TJ/F11 9.9626 Tf 10.694 0 Td [(pt = O p p .Themeanandvarianceofthebinomial randomvariable B )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m + p +1 ;s= t + s are,respectively, = s m + p +1 t + s ; and 2 = st m + p +1 t + s 2 : Normalizing,wehave B m + p +1 ; s t + s ;p = P B )]TJ/F11 9.9626 Tf 9.963 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [( : 66

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If ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 ,with < 1 ,then p )]TJ/F11 9.9626 Tf 8.564 0 Td [( = p )]TJ/F1 9.9626 Tf 8.564 14.047 Td [( p + h + s t + s = )]TJ/F11 9.9626 Tf 7.749 0 Td [(h )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ o ; and = s t p + h t + s + st t + s 2 = r pt t + s )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ o : Therefore, p )]TJ/F17 7.9701 Tf 6.586 0 Td [( = )]TJ/F10 6.9738 Tf 6.226 0 Td [( p t t + s )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ o ,anditfollowsfrom.3thatas p !1 .4 1 2 t + s m + p +1 s p t m B m + p +1 ; s t + s ;p 1 2 t + s m + p +1 s p t m )]TJ/F11 9.9626 Tf 7.749 0 Td [( p t t + s : Wenowgivetheasymptoticformof.2with ms )]TJ/F11 9.9626 Tf 10.304 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 < 1 ,whichwillalsobethe asymptoticformof G m;p ; s;t withthosesameconditions. Lemma5.3. Supposethatas p !1 ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 < 1 .Thenasymptotically .2 equals p 2 pt t + s 2 exp 2 2 t t + s )]TJ/F11 9.9626 Tf 7.748 0 Td [( p t t + s : Proof. Combining.4withtheresultofLemma5.2,.2equals p 2 pt t + s 2 )]TJ/F11 9.9626 Tf 7.748 0 Td [( p t t + s 1+ h p )]TJ/F10 6.9738 Tf 6.226 0 Td [(p 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m )]TJ/F10 6.9738 Tf 6.227 0 Td [(m )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o : Thuswearelefttoshow .5 1+ h p )]TJ/F10 6.9738 Tf 6.226 0 Td [(p 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m )]TJ/F10 6.9738 Tf 6.227 0 Td [(m exp 2 2 t t + s : Thefollowinginequalitiesholdforall x 0: 1+ x exp x )]TJ/F11 9.9626 Tf 11.159 6.739 Td [(x 2 2 ; 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F11 9.9626 Tf 11.158 6.739 Td [(x 2 2 : Then 1+ h p p exp h )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(h 2 2 p ; 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m m exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(h )]TJ/F11 9.9626 Tf 12.918 6.74 Td [(h 2 2 m : Also,given > 0andsucientlysmall x wehave 1+ x exp x )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(x 2 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [( ; 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(x 2 2 + : Since h = o p and h = o m ,forany > 0thereisa M suchthatmin f m;p g >M implies 1+ h p p exp h )]TJ/F11 9.9626 Tf 11.158 6.739 Td [(h 2 2 p )]TJ/F11 9.9626 Tf 9.962 0 Td [( ; 1 )]TJ/F11 9.9626 Tf 12.662 6.739 Td [(h m m exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(h )]TJ/F11 9.9626 Tf 12.918 6.739 Td [(h 2 2 m + : 67

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Thenforsucientlylarge p wehave exp )]TJ/F11 9.9626 Tf 8.945 6.74 Td [(h 2 2 1 m + 1 p + 1+ h p )]TJ/F10 6.9738 Tf 6.227 0 Td [(p 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m )]TJ/F10 6.9738 Tf 6.227 0 Td [(m exp )]TJ/F11 9.9626 Tf 8.944 6.74 Td [(h 2 2 1 m + 1 p + : Now.5followssince h 2 2 1 m + 1 p = 1 2 h 2 p m + p m 1 2 2 t + s 2 t + s t = 2 2 t t + s ; andwecanset tobeassmallaswewish.Therefore,wehavecompletedtheproof. If p p = o ms )]TJ/F11 9.9626 Tf 10.468 0 Td [(pt ,weareintrouble,aswewillhave .Normalapproximationwill thusfailus,soanotherapproachisneeded.Thefollowingtheoremswilltakecareofourremaining cases.Sinceweareunabletondproofsintheliterature,weshallprovideproofshere.Theproofof Lemma5.4isprovidedbytheanonymousrefereeto[21],andwillbeusedforthecasewith p p = o ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt and ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt = o p Lemma5.4. Assumethat m nq where q candependon n andlet r =1 )]TJ/F11 9.9626 Tf 10.081 0 Td [(q and h = nq )]TJ/F11 9.9626 Tf 10.081 0 Td [(m Supposethatthefollowingconditionsholdas n !1 : C1 nqr=h 2 0 C2 nqr=h !1 Thenas n !1 wehave B n;q;m m n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m nh b n;q;m : Thatis, B n;q;m = X 0 k m n k q k r n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k nqr h n m q m r n )]TJ/F10 6.9738 Tf 6.227 0 Td [(m : Proof. Notethat,forall n q k ,wehave b n;q;k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 b n;q;k = kr q n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k +1 : Thisratioisboundedaboveby x = mr=q n )]TJ/F11 9.9626 Tf 8.618 0 Td [(m +1when k m .Thus b n;q;m )]TJ/F11 9.9626 Tf 8.618 0 Td [(j b n;q;m x j Wehave x = 1 )]TJ/F11 9.9626 Tf 13.681 6.739 Td [(h nq 1+ h +1 nr =1 )]TJ/F11 9.9626 Tf 17.262 19.396 Td [(h nqr + 1 nr 1+ h +1 nr ; 68

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andthus1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x h=nqr .Therefore, B n;q;m b n;q;m + x + x 2 + = b n;q;m 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x b n;q;m nqr h : Notealsothatif k m )]TJ/F11 9.9626 Tf 9.963 0 Td [(z then kr q n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k +1 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(t r q n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + t +1 ; thelatterweshallcall y .Thenfor j z wehave b n;q;m )]TJ/F11 9.9626 Tf 9.962 0 Td [(j b n;q;m y j .With y = 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( h + t nq 1+ h + t +1 nr =1 )]TJ/F11 9.9626 Tf 16.852 19.397 Td [(h + t nqr + 1 nr 1+ h + t +1 nr ; wechoose z sothat z = o h and zt=nqr !1 ,then1 )]TJ/F11 9.9626 Tf 8.997 0 Td [(y h=nqr and y t = o .Thusweconclude that B n;q;m b n;q;m + y + + y t = b n;q;m 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y t +1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y b n;q;m nqr h : Lemma5.5. Supposethatas p !1 ms )]TJ/F11 9.9626 Tf 11.073 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 !1 and ms )]TJ/F11 9.9626 Tf 11.073 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0 .Then asymptotically .2 equals t t + s 2 p ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt : Proof. Withtheconditions ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 !1 and ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 0,wehave st m + p +1 ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt + s 0 ; st t + s m + p +1 ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + s !1 ; soconditionsC1andC2ofLemma5.4aresatised.Therefore, B m + p +1 ; s t + s ;p st m + p +1 t + s ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + s m + p +1 p s t + s p t t + s m +1 ; and.2thusasymptotically t 2 m + p +1 st t + s ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + s m + p +1 m +1 t t + s 2 p ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt ; since m + p m t + s t and m + p p t + s s If ms )]TJ/F11 9.9626 Tf 10.173 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 tendstoaconstantgreaterthanzero,thatis, ms=pt > 1,thenwecannot transitionfromthecrossingsformof G m;p ; s;t to.2withoutsignicanterror,outsideofthe 69

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case s =1,wherewehavethebinomialform.Thus,theresultbelowweshallonlyapplywhen s =1. Lemma5.6. Let m = nq )]TJ/F11 9.9626 Tf 9.646 0 Td [(h and h = vnqr where v candependon n .Assumethatthefollowing conditionsaresatised.As n !1 C3 m !1 ;n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m !1 C4 v> forsomeconstant > 0 Thenas n !1 B n;q;m 1+ qv v b n;q;m : Proof. Let t k = b n;p;k = n k p k q n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ;r k = t k t k )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : Then r k = n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k +1 p kq : For k m ,wehave r k n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m +1 p mq n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m p mq = nq + h p np )]TJ/F11 9.9626 Tf 9.963 0 Td [(h q = 1+ pv 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(qv : Wewrite r = 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(qv 1+ pv = mq n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m p : Notethat .6 r =1 )]TJ/F11 9.9626 Tf 22.243 6.74 Td [(v 1+ pv 1 )]TJ/F11 9.9626 Tf 19.737 6.74 Td [(v 1+ v = 1 1+ v < 1 1+ < 1 ; wherethesecondtolastinequalityholdsforalllarge n .Thus,for k m ,wehave .7 r k 1 =r> 1 : Wethereforehavefor k =1 ; 2 ;:::;m that .8 t m )]TJ/F10 6.9738 Tf 6.226 0 Td [(k r k t m : Hence, B n;p;m = m X k =0 t m )]TJ/F10 6.9738 Tf 6.226 0 Td [(k m X k =0 r k t m t m 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(r = t m 1+ pv v : 70

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Togetthecorrespondingasymptoticlowerbound,welet besuchthatas n !1 satisesthe following: !1 ; = o n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m ; and = o m Since B n;p;m = m X k =0 t k X m )]TJ/F10 6.9738 Tf 6.227 0 Td [( k m t k ; Itissucienttoshowthatas n !1 .9 X m )]TJ/F10 6.9738 Tf 6.227 0 Td [( k m t k t m 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(r : For m )]TJ/F11 9.9626 Tf 9.962 0 Td [( k m ,wehave r k = n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k +1 p kq n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + +1 p m )]TJ/F11 9.9626 Tf 9.963 0 Td [( q = n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m p mq 0 B @ 1+ +1 n )]TJ/F11 9.9626 Tf 9.962 0 Td [(m 1 )]TJ/F11 9.9626 Tf 12.88 6.74 Td [( m 1 C A : Wewrite R = r 0 B @ 1 )]TJ/F11 9.9626 Tf 12.879 6.74 Td [( m 1+ +1 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m 1 C A : Wenotethatsince =m = o and +1 = n )]TJ/F11 9.9626 Tf 9.55 0 Td [(m = o ,wehave R r ,as n !1 .Also, R< 1 foralllarge n becauseof.6.Therefore,since t m )]TJ/F10 6.9738 Tf 6.226 0 Td [(k R k t m ; 0 k ; wehaveas n !1 that X m )]TJ/F10 6.9738 Tf 6.227 0 Td [( k m t k X 0 k R k t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(R t m 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(R t m 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(r : Thisshows.9andourproofiscomplete. Lemma5.7. Supposethat,as p !1 ms=pt > 1 .Then .2 isasymptotically t 2 + t + s 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Proof. With v = t + s st ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + s m + p +1 t + s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 t + s > 0 ; 71

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conditionsC3andC4ofLemma5.6aremet,andtherefore B m + p +1 ; 1 t +1 ;p t + s t + s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 1+ s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t + s m + p +1 p t m +1 s p t + s m + p +1 = )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m + p +1 p t m +1 s p t + s m + p +1 : Then.2becomes m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m + p +1 p = t 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m + p +1 m +1 t + s 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = t 2 + t + s 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : 5.2. TransitioningtothePseudo-BinomialForm. Inthissection,wegiveageneraloutlineon howweshallre-moldthedouble-doublezero-betsumformula m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 M i X k =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i .10 + t X j =1 M )]TJ/F10 6.9738 Tf 6.227 0 Td [(j X k =0 j k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 toward.2.Weshalldothisviatheintermediary .11 m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p X k =0 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ; whichequalsthesum m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t 2 p X k =0 m + p +1 k t s p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k byTheorem3.1.Partofthistransitioninvolvesatailexchange.Let A bealargeconstant,and dene T A = f k :0 k A or p=s )]TJ/F11 9.9626 Tf 9.962 0 Td [(A k p=s g ; T A = f k :0 k As or p )]TJ/F11 9.9626 Tf 9.963 0 Td [(As k p g : Weshallassumeabitofexibilityontheupperend-betheterm p=s "itshouldbeunderstoodto actuallymean M i ."Weshowthatthecontributionofthedouble-doublesumsover T A andthe singlesumover T A canbereasonablybounded. Lemma5.8. Let A bealargeconstant.Thenforeach i 0 i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 and 1 j t wehave m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 T A k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i = O A 72

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and m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k 2 T A k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = O A : Proof. Thisfollowssinceboth m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i and m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 areprobabilities,andweareremovingatmost2 A termsforeach. Sincethenumberofsuchsumsisdependentonlyontheconstants s and t ,thesumofallthe tailsover i and j arestill O A .Thereplacementtailis m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t 2 X k 2 T A k t + s s k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : Weprovethatthistailalsoissimilarlyboundedintermsof A : Lemma5.9. Let A bealargeconstant.Then m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k 2 T A k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = O A : Proof. Forthissum,let a and b bearbitrarynonnegativerealnumbersandconsiderthesequence a + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k b + k k ; 0 k m: Thissequenceiseitherincreasingordecreasing,dependingonwhether a>b ornot.Thus,forany nonnegativerealnumbers a b ,wehave a + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k b + k k max a + p p ; b + p p a + b + p p : Taking a = m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt=s b = kt=s ,wehavethat m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k 1 for0 k p .Thusthesumover T A is O A 73

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Thedominantpartofthedouble-doublesumslieswithintheremainingrangeof k .Further trimmingofthesecondarytail"shallbedealtwithcasebycase. Oncewehaveidentiedthedominantrangeofthedouble-doublesumformula.10,webegin worktoreducethatsumto.2.Webeginbyfolding"thetwodoublesumstogether.Thenwe shallbeaslightadjustmentawayfrombeingabletorewritetheresultingdoublesumintermsof thepseudo-binomialform.2. Weshallfoldthetwodoublesumstogetherbyassertingthatforsmallconstants a and b with 0 a + b s + t ,underappropriateconditionswehave k t + s + a + b ks + b m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b k t + s ks m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks : Ourgoalisnottoridourselvesoftheconstants a i b i and c i ,rather,wewanttoexchangethose withinthedown-crossingsums j< 0withanothersetcorrespondingtoanup-crossingsum i 0. Inthiswayweshallfold"thedown-crossingssumintotheup-crossingssum. Wehave k t + s + a + b ks + b = k t + s ks )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k t + s + a + b a + b kt + a a ks + b b .12 k t + s ks )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( k +1 t + s a + b ks b kt a = k t + s ks t + s s b t + s t a 1+ 1 k a + b : Similarly,recallingthat b s and a t k t + s + a + b ks + b k t + s ks )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k t + s a + b )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( k +1 s b )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( k +1 t a .13 = k t + s ks t + s s b t + s t a 1 )]TJ/F8 9.9626 Tf 19.997 6.74 Td [(1 k +1 a + b : Weworksimilarlywiththeotherbinomialcoecients. m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b = m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks b m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt a )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s a + b m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks b m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt a )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k +1 t + s a + b : Writing ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt = h t + s ,wehave m + p )]TJ/F8 9.9626 Tf 8.856 0 Td [( k +1 t + s p )]TJ/F11 9.9626 Tf 8.855 0 Td [(ks = t + s s 1+ h )]TJ/F11 9.9626 Tf 8.856 0 Td [(s p )]TJ/F11 9.9626 Tf 8.856 0 Td [(ks and m + p )]TJ/F8 9.9626 Tf 8.856 0 Td [( k +1 t + s m )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt = t + s t 1 )]TJ/F11 9.9626 Tf 14.305 6.74 Td [(h + t m )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt : 74

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Therefore p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks b m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt a )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k +1 t + s a + b = t + s s )]TJ/F10 6.9738 Tf 6.227 0 Td [(b t + s t )]TJ/F10 6.9738 Tf 6.226 0 Td [(a 1+ h )]TJ/F11 9.9626 Tf 9.962 0 Td [(s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b 1 )]TJ/F11 9.9626 Tf 15.411 6.74 Td [(h + t m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a : Thus m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b .14 m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks t + s s )]TJ/F10 6.9738 Tf 6.226 0 Td [(b t + s t )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 1+ h )]TJ/F11 9.9626 Tf 9.963 0 Td [(s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b 1 )]TJ/F11 9.9626 Tf 15.412 6.74 Td [(h + t m + kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a : Similarly,wecanshowthat m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k +1 s b )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m )]TJ/F8 9.9626 Tf 9.962 0 Td [( k +1 t a )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s a + b .15 = m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks t + s s )]TJ/F10 6.9738 Tf 6.227 0 Td [(b t + s t )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 1+ h + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(s )]TJ/F10 6.9738 Tf 6.226 0 Td [(b 1 )]TJ/F11 9.9626 Tf 23.299 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(t )]TJ/F10 6.9738 Tf 6.227 0 Td [(a : Dene .16 U 1 m;p;s;t = 1+ 1 k a + b 1+ h )]TJ/F11 9.9626 Tf 9.963 0 Td [(s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b 1 )]TJ/F11 9.9626 Tf 15.412 6.74 Td [(h + t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a and .17 L 1 m;p;s;t = 1 )]TJ/F8 9.9626 Tf 19.996 6.74 Td [(1 k +1 a + b 1+ h + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(s )]TJ/F10 6.9738 Tf 6.226 0 Td [(b 1 )]TJ/F11 9.9626 Tf 23.299 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(t )]TJ/F10 6.9738 Tf 6.226 0 Td [(a ; sothatforany i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i k t + s ks m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks U 1 m;p;s;t .18 and k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i k t + s ks m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks L 1 m;p;s;t : .19 75

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Thisimpliesforany )]TJ/F11 9.9626 Tf 7.749 0 Td [(t i;j s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1: k t + s + c j ks + b j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b j L 1 m;p;s;t U 1 m;p;s;t .20 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i k t + s + c j ks + b j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b j U 1 m;p;s;t L 1 m;p;s;t : Undertherightconditions,wecanmarginalize U 1 and L 1 .Inthiswayweshallbeabletofold" thetwodoublesumsintoasingledoublesum.Asanaddedbonus,wecandoitinawaythatwill leaveonlythetwobinomialtermsinsidethesum. With.10nowreducedtoadoublesum,ournextstepistoreindextheinnersumwithrespect to b i ,usingLemma4.9.Withthemap asdenedinLemma4.9,if i = b i = j ,thenfrom i = b i t )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i s = jt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i s wehave a i = jt )]TJ/F11 9.9626 Tf 9.963 0 Td [(i =s .Therefore, s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 M i X k =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i .21 = s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 M i X k =0 k + j s t + s )]TJ/F10 6.9738 Tf 11.628 3.922 Td [(i s ks + j m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j s t + s + i s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j ; where i = b i = j .Nowwewanttoremovethetwo i=s terms.Whilebasicallyatrade"between thetwobinomialcoecientslikethepreviousexchange,herewehavetochangeeveryfactorinstead ofdiscardingacouple.Sincewehavebottomentriesnotdependenton i ,wecanplacethemto thesideandworkwiththeremainingfallingfactorials.Todrawacomparison,weshallusethe harmonicnumbers.The n th harmonicnumberis .22 H n = n X k =1 1 k : Asymptotically,forlarge n wehave H n =ln n + + o =n ; where 0 : 5772isEuler'sconstant.Thusforlarge a wehave b X k = a +1 1 k = H b )]TJ/F11 9.9626 Tf 9.963 0 Td [(H a =ln b a + o =n : 76

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Webeginwith k + j s t + s )]TJ/F11 9.9626 Tf 11.777 6.74 Td [(i s ks + j and k + j s t + s ks + j : Wehaveanaturalpairingofthefactors.Takingagenericfactorpair,wehave k + j=s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i=s )]TJ/F11 9.9626 Tf 9.963 0 Td [(l k + j=s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(l =1 )]TJ/F11 9.9626 Tf 46.26 6.74 Td [(i=s k + j=s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(l exp )]TJ/F11 9.9626 Tf 44.047 6.74 Td [(i=s k + j=s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(l exp )]TJ/F11 9.9626 Tf 68.974 6.739 Td [(i=s k + j=s t + s + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i =s )]TJ/F11 9.9626 Tf 9.963 0 Td [(l ; sincefor1 i s )]TJ/F8 9.9626 Tf 10.37 0 Td [(1, k + j=s t + s isnotaninteger )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( k + j=s t + s )]TJ/F11 9.9626 Tf 10.37 0 Td [(i=s is .Takingthe productover0 l ks + j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1,wehave )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( k + j=s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i=s ks + j )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( k + j=s t + s ks + j .23 exp )]TJ/F11 9.9626 Tf 9.563 6.74 Td [(i s H k + j s t + s + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H k + j s t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s !# : Asfortheothersetoffallingfactorials, )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j=s t + s + i=s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j=s t + s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(j .24 exp i s H m + p )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t + s )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H m )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s !# ; Wecandevelopcorrespondinglowerboundsbyusingthesamemethodonthereciprocals. Withfavorablecircumstanceswewillbeabletoerasethe i=s termswithoutintroducingany signicanterror.Wecompletethetransitiontoasinglesumbynotingthat p X k =0 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 M i X k =0 k + j=s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j=s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j .25 bysplittingthesumaccordingtotheresidueof k modulo s .Now,from.11weshiftto.2. Thiscompletesthetransitiontothepseudo-binomialform.Now,weshallgotoacase-by-case analysis,whereweshallllinthegaps. 77

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5.3. Urnswith pt )]TJ/F11 9.9626 Tf 9.709 0 Td [(ms = O p p Webeginwiththeurnsforwhich pt )]TJ/F11 9.9626 Tf 9.709 0 Td [(ms = O p p as p !1 Theorem5.10. Supposethat,as p !1 ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt = p p ,with j j < 1 .Then,as p !1 G m;p ; s;t max f)]TJ/F11 9.9626 Tf 12.731 0 Td [( p p; 0 g + p 2 pt t + s 2 exp 2 2 t t + s j j p t t + s : Inparticular,if pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms = o p p ,wehave G m;p ; s;t 1 4 p 2 pt t + s Proof. Weshallprovetheformularstfor pt )]TJ/F11 9.9626 Tf 10.176 0 Td [(ms 0.Then,weshallusetheAntiurnTheorem toshowthattheresultalsoholdswhen pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms> 0.Deneforconstants A> 0and 2 ; 1: U A; = n k : )]TJ/F11 9.9626 Tf 9.963 0 Td [( p s k p s )]TJ/F11 9.9626 Tf 9.963 0 Td [(A o ; U A; = f k : )]TJ/F11 9.9626 Tf 9.963 0 Td [( p k p )]TJ/F11 9.9626 Tf 9.963 0 Td [(As g D A; = n k : A k )]TJ/F11 9.9626 Tf 9.963 0 Td [( p s o ; D A; = f k : As k )]TJ/F11 9.9626 Tf 9.962 0 Td [( p g With pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms 0,wehave m + p p G m;p ; s;t = s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 M i X k =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i + t X j =1 M )]TJ/F10 6.9738 Tf 6.227 0 Td [(j X k =0 j k t + s + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Weassertthatwecaneectivelyreplacetheportionofthesum.10forwhich k 2 D A; withthe portionof.11with k 2 D A; .Indeed,letting ms )]TJ/F11 9.9626 Tf 9.772 0 Td [(pt = h t + s ,from.16and.17wehave thatfor k 2 D A; that U 1 m;p ; s;t =1+ O =A and L 1 m;p ; s;t =1+ O =A .Thusforany 1 j t and0 i s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1and k 2 D A; : k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j m + p )]TJ/F11 9.9626 Tf 8.856 0 Td [(k t + s )]TJ/F11 9.9626 Tf 8.856 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j p )]TJ/F11 9.9626 Tf 8.855 0 Td [(ks )]TJ/F11 9.9626 Tf 8.856 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j = )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 8.856 0 Td [(k t + s )]TJ/F11 9.9626 Tf 8.856 0 Td [(c i p )]TJ/F11 9.9626 Tf 8.855 0 Td [(ks )]TJ/F11 9.9626 Tf 8.856 0 Td [(b i : Removingthefactors m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i = t + s t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o and m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j = t + s s )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ o ; wehave s k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A t k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i : 78

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Since s t + s t X j =1 j + t t + s s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 i = 1 t + s s t t +1 2 + t s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 s 2 = st 2 ; .26 over D A; wehaveforeachsuch k that s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i + t X j =0 j k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A t 2 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i ; sincewearesplitting st= 2into s parts.Reindexingwithrespectto b i = j byLemma4.9,werewrite thesum s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i asthesum s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k + j s t + s )]TJ/F10 6.9738 Tf 11.629 3.923 Td [(i s ks + j m + p )]TJ/F1 9.9626 Tf 9.963 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k + j s t + s + i s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j : Since k A ,wehave H k + j s t + s + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H k + j s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s = O =A +ln k + j s t + s + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s k + j s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s = O =A + )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ o ln t + s t : Thusfrom.23wehave exp )]TJ/F11 9.9626 Tf 9.563 6.74 Td [(i s H k + j s t + s + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H k + j s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s !# .27 =exp )]TJ/F11 9.9626 Tf 9.564 6.74 Td [(i s O =A + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o ln t + s t = t + s t )]TJ/F10 6.9738 Tf 6.227 0 Td [(i=s )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A ; 79

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withthesameresultfromthelowerbound.Asfortheotherset,since k )]TJ/F11 9.9626 Tf 9.963 0 Td [( p=s wehave H m + p )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t + s )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H m )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s =ln t + s p + h )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j t p + h )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j + h + O =p =ln t + s t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A + O =p : Exponentiating,from.24wethushave exp i s H m + p )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t + s )]TJ/F11 9.9626 Tf 11.158 6.739 Td [(s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H m )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t )]TJ/F11 9.9626 Tf 11.158 6.739 Td [(s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s !# .28 =exp i s ln t + s t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A + O =p = t + s t i=s )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A : Equations.27and.28implythatfor k 2 D A; ,wehave k + j s t + s )]TJ/F10 6.9738 Tf 11.629 3.923 Td [(i s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s + i s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j = k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A : Wehavethusshownthatfor k 2 D A; s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i .29 + t X j =1 j k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ O =A t 2 s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j : Thesum m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 X k 2 D A; k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j equals O + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 D A; k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ; 80

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uponsplittingthelattersumaccordingtotheresidueof k modulo s .Weshallhidethe O term inthe O A termcreatedbythetails.Asforthosetails,Lemma5.8takescareoftheextremeends k 2 T A k 2 T A .Theremainingrangeisthesecondarytail,"thesumover U A; = n k : )]TJ/F11 9.9626 Tf 9.963 0 Td [( p s k p s )]TJ/F11 9.9626 Tf 9.962 0 Td [(A o : Todealwiththis,weshallboundfromabovethecontributionsofthetermswith k 2 U A; interms of and A .As p !1 ,weshallbeabletolet 0,whichwillmakethesebounds,hencethe contributions,tendtozero.Weshallreplacethissectionwiththeportionofthesum.11forwhich k 2 D A; .Then,wewillhavecompletedthetransitionfrom G m;p ; s;t tothepseudo-binomial form.2. Lemma5.11. Suppose ms )]TJ/F11 9.9626 Tf 8.813 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 < 1 ,with > 0 ,as p !1 .Let A bealargeconstant, andlet 2 ; 1 .Then,as p !1 thereisaconstant c 1 ,with c 1 0 as 0 ,suchthatthe sumof m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 U A; k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i over 1 i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 and m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 U A; k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 over 1 j t isboundedaboveby c 1 p p Proof. WeapplyStirling'sformulatothevariousbinomialcoecients.Write ms )]TJ/F11 9.9626 Tf 10.219 0 Td [(pt = h t + s ByLemma5.2wehave m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = t + s s )]TJ/F10 6.9738 Tf 6.227 0 Td [(p t + s t )]TJ/F10 6.9738 Tf 6.227 0 Td [(m r 2 mp m + p 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m )]TJ/F10 6.9738 Tf 6.227 0 Td [(m 1+ h p )]TJ/F10 6.9738 Tf 6.227 0 Td [(p )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A : .30 Wehaveforany )]TJ/F11 9.9626 Tf 7.749 0 Td [(t i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1that k t + s + c i ks + b i = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A s + t s ks + b i s + t t kt + a i r t + s 2 kts : .31 Let d i = s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i = t + s for1 i s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1.Thenwehaverecallingthat tb i )]TJ/F11 9.9626 Tf 9.962 0 Td [(sa i = i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i = t + s s 1+ h )]TJ/F11 9.9626 Tf 9.962 0 Td [(d i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i 81

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and m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = t + s t 1 )]TJ/F11 9.9626 Tf 31.918 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.962 0 Td [(d i m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Let e j = j )]TJ/F11 9.9626 Tf 9.963 0 Td [(t = t + s for1 j t .Thenwealsohave m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = t + s s 1+ h )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 and m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j = t + s t 1 )]TJ/F11 9.9626 Tf 26.726 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j : Then m + p )]TJ/F11 9.9626 Tf 8.856 0 Td [(k t + s )]TJ/F11 9.9626 Tf 8.856 0 Td [(c i )]TJ/F8 9.9626 Tf 8.856 0 Td [(1 p )]TJ/F11 9.9626 Tf 8.855 0 Td [(ks )]TJ/F11 9.9626 Tf 8.855 0 Td [(b i = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A s m + p )]TJ/F11 9.9626 Tf 8.856 0 Td [(k t + s )]TJ/F11 9.9626 Tf 8.855 0 Td [(c i )]TJ/F8 9.9626 Tf 8.855 0 Td [(1 2 p )]TJ/F11 9.9626 Tf 8.856 0 Td [(ks )]TJ/F11 9.9626 Tf 8.856 0 Td [(b i m )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt )]TJ/F11 9.9626 Tf 8.855 0 Td [(a i )]TJ/F8 9.9626 Tf 8.856 0 Td [(1 .32 t + s s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b i t + s t m )]TJ/F10 6.9738 Tf 6.227 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1+ h )]TJ/F11 9.9626 Tf 8.855 0 Td [(d i p )]TJ/F11 9.9626 Tf 8.856 0 Td [(ks )]TJ/F11 9.9626 Tf 8.856 0 Td [(b i p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b i 1 )]TJ/F11 9.9626 Tf 28.597 6.739 Td [(h )]TJ/F11 9.9626 Tf 8.856 0 Td [(d i m )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt )]TJ/F11 9.9626 Tf 8.856 0 Td [(a i )]TJ/F8 9.9626 Tf 8.855 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.226 0 Td [(a i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ; and m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A t + s s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b )]TJ/F9 4.9813 Tf 5.396 0 Td [(j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t + s t m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F9 4.9813 Tf 5.396 0 Td [(j .33 1+ h )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b )]TJ/F9 4.9813 Tf 5.396 0 Td [(j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 )]TJ/F11 9.9626 Tf 26.726 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m )]TJ/F10 6.9738 Tf 6.227 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F9 4.9813 Tf 5.397 0 Td [(j s m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j : Noticethatinthecase1 i s )]TJ/F8 9.9626 Tf 10.039 0 Td [(1,allpowersof t + s =s from.30,5.31,and.32cancel, withthe t + s =t factorsreducedtoalone t= t + s .When1 j t ,.30,.31,and.33has allpowersof t + s =t canceling,andalonefactorof s= t + s remaining. Wenowfocusontheplus"andminus"terms.Sincethefunction+1 =x x isincreasing in x ,wehave 1+ h )]TJ/F11 9.9626 Tf 9.962 0 Td [(d i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b i 1+ h p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i p )]TJ/F10 6.9738 Tf 6.226 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b i 1+ h p p ; .34 1+ h )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b )]TJ/F9 4.9813 Tf 5.396 0 Td [(j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1+ h p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b )]TJ/F9 4.9813 Tf 5.396 0 Td [(j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1+ h p p : .35 82

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Since )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =x x isalsoincreasingin x ,wehave 1 )]TJ/F11 9.9626 Tf 31.918 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(d i m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.227 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(d i m m = 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m m + o ; .36 1 )]TJ/F11 9.9626 Tf 26.726 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.962 0 Td [(e j m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F9 4.9813 Tf 5.397 0 Td [(j 1 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(h )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j m m = 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m m )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o ; .37 sinceboth d i and e j are o m .From.30,.31,.32,.34,and.36weconcludethat m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i .38 s mp t + s m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 kts m + p p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A t t + s ; while.30,.31,.33,.35,and.37giveus m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k t + s + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .39 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A s t + s s mp t + s m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 kts m + p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ; Observethat m m + p t t + s ; m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t + s t ; and p k ps )]TJ/F11 9.9626 Tf 9.962 0 Td [( p = s 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( : Hence.38isboundedaboveby )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A t t + s s t + s 2 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i : Nowwesumover k .Concentratingonthetermdependenton k : b p=s c)]TJ/F10 6.9738 Tf 9.879 0 Td [(A X k = )]TJ/F10 6.9738 Tf 6.226 0 Td [( p=s 1 p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i b p=s c)]TJ/F10 6.9738 Tf 9.88 0 Td [(A +1 X k = )]TJ/F10 6.9738 Tf 6.227 0 Td [( p=s +1 1 p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks .40 1 p s p=s +1 X k =1 1 p k 2 r p=s +1 s = 2 p p s )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o : Thusforeach i ,thesumof.38over k isboundedaboveby )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ O =A 2 t s t + s s p t + s 2 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( ; 83

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thusthecontributionoftherstdoublesuminthestatementofthelemmaisboundedaboveby )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A t s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t + s s p t + s 2 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( : .41 Workingsimilarlywith.39,foreach j thesumover k isboundedaboveby )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A 2 t + s s p t + s 2 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( ; andthecontributionoftheseconddoublesumisboundedaboveby .42 )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(1+ O =A t t +1 s + t s p t + s 2 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( : Thuswetake c 1 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A s t t + s 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [( : Wewanttoreplacethisportionofthesumwith X k 2 D A; k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Lemma5.12. Suppose ms )]TJ/F11 9.9626 Tf 8.813 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 < 1 ,with > 0 ,as p !1 .Let A bealargeconstant, andlet 2 ; 1 .Then,as p !1 thereisaconstant c 2 ,with c 2 0 as 0 ,suchthat m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 D A; k t + s s k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k c 2 p p; with c 2 0 as 0 Proof. Wewrite )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(with i denedsothat i = j m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k 2 D A; k t + s s k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = O + m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 X k 2 U A; k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j O + m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 X k 2 U A; k + j s t + s + s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i s ks + j m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j s t + s + i s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j = O + m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 X k 2 U A; k t + s + c i +1 ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i ; 84

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andsince k islargeenough, k t + s + c i +1 kt + a i +1 = t + s t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o reducesbacktoLemma5.11afterdivisionby )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p Letting C = c 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 2 ,wehaveshownthatfor p largeenoughand ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt 0, G m;p ; s;t = O A + C p p + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t 2 p X k =0 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k .43 = O A + C p p + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 t 2 p X k =0 m + p +1 k t s p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k ; byLemma5.8.ApplyingLemma5.3,thenletting A !1 and 0,weconcludethat G m;p ; s;t p 2 pt t + s 2 exp 2 2 t t + s )]TJ/F11 9.9626 Tf 7.748 0 Td [( p t t + s ; asdesired. If pt )]TJ/F11 9.9626 Tf 9.844 0 Td [(ms 0,thenfromtheAntiurnTheoremwehave G m;p ; s;t = G p;m ; t;s + pt )]TJ/F11 9.9626 Tf 9.844 0 Td [(ms : Then,since ms pt ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt p p 0 pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms p m !)]TJ/F11 9.9626 Tf 20.479 0 Td [( r t s 0 : Thusbyourpreviousresult, G p;m ; t;s p 2 ms t + s 2 exp 2 s=t 2 s t + s p s=t p s t + s p 2 pt t + s 2 exp 2 2 t t + s p t t + s : Therefore,when pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms 0wehave G m;p ; s;t )]TJ/F11 9.9626 Tf 18.265 0 Td [( p p + p 2 pt t + s 2 exp 2 2 t t + s j j p t t + s : If pt )]TJ/F11 9.9626 Tf 10.277 0 Td [(ms = o p p ,thenwehave =0.Then,sinceexp=1and=1 = 2,thesecondary resultholds. 85

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Corollary5.13. TheresultofTheorem5.10holdsforanypositive s and t Proof. Supposerstthat t=s isrational.Wecanwrite G m;p ; s;t = nG m;p ; r;q ,where q=r is t=s writteninlowestterms.Thatis, nr = s and nt = q .Then,if pt )]TJ/F11 9.9626 Tf 10.254 0 Td [(ms = p p as p !1 then pq )]TJ/F17 7.9701 Tf 6.586 0 Td [(mr p p n .ThenbyTheorem5.10wehave G m;p ; s;t n p 2 pq q + r 2 exp =n 2 2 q q + r j =n j p q q + r = p 2 pnq nq + nr 2 exp 2 2 nq nq + nr j j p nq nq + nr = p 2 pt t + s 2 exp 2 2 t t + s j j p t t + s : Theextensiontothecasewith t=s irrationalfollowsbythecontinuityof G m;p ; s;t in s and t 5.4. Urnswith pt )]TJ/F11 9.9626 Tf 10.216 0 Td [(ms 0 p p = o pt )]TJ/F11 9.9626 Tf 10.216 0 Td [(ms ,and pt )]TJ/F11 9.9626 Tf 10.215 0 Td [(ms = o p Wenextexamineurnsfor which ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt = p p !1 ,but ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt =p 0.Wendthatforlarge p ,weshouldnotexpectto gainasmuchasanurnwithaninitialweightclosertozero.Thisisintuitivelyclear.Theurnhas alongertriptoneutral,andthuswilltakealongeramountoftimetoneutralize.Thereforethere willbelesstimetocollectgains. Theorem5.14. Supposethatas p !1 ms )]TJ/F11 9.9626 Tf 10.201 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 !1 and ms )]TJ/F11 9.9626 Tf 10.201 0 Td [(pt p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 0 .Thenfor sucientlylarge p G m;p ; s;t t t + s 2 p ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt : Proof. AsintheproofofTheorem5.10,wewanttoreplacethedouble-doublesumfoundinthe crossingsform, s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 M i X k =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i + t X j =1 M )]TJ/F10 6.9738 Tf 6.227 0 Td [(j X k =0 j k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ; withthesinglesum t 2 p X k =0 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k = t + s m + p +1 2 s p t m B m + p +1 ; s t + s ;p : 86

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Theprocessissimilarinmanyrespects.Write ms )]TJ/F11 9.9626 Tf 10.077 0 Td [(pt = h t + s ,dene T A and T A asbefore, andfor !> 0let U A;! = n k : p s )]TJ/F11 9.9626 Tf 9.962 0 Td [(!h k p s )]TJ/F11 9.9626 Tf 9.963 0 Td [(A o ; U A;! = f k : p )]TJ/F11 9.9626 Tf 9.963 0 Td [(!hs k p )]TJ/F11 9.9626 Tf 9.963 0 Td [(As g ; D A;! = n k : A k p s )]TJ/F11 9.9626 Tf 9.962 0 Td [(!h o ; D A;! = f k : As k p )]TJ/F11 9.9626 Tf 9.962 0 Td [(!hs g : Lemmas5.8and5.9willallowustoswapthetails.Thesecondarytails,with k inrespectively U A;! or U A;! ,shallbedealtwithlater.Thedominantportionofthedouble-doublesumhas k in D A;! .ApplyingthesameprocessasinTheorem5.10,from.20wehavefor k 2 D A;! and 0 i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1,1 j t : U 1 m;p ; s;t L 1 m;p ; s;t = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.132 -8.07 Td [(1+ O =! : With m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i = t + s t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o ; and m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j = t + s s )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ o ; from.26wecansaythatfor k 2 D A;! : s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i + t X j =1 j k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.132 -8.07 Td [(1+ O =! t 2 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i ; whereagainwesplit st= 2into s parts.Reindexingwithrespectto b i ,thesumbecomes .44 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 k + j s t + s )]TJ/F10 6.9738 Tf 11.629 3.922 Td [(i s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s + i s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j : Wenowremovethe i=s terms.From.23wehave exp )]TJ/F11 9.9626 Tf 9.563 6.74 Td [(i s H k + j s t + s + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s )]TJ/F11 9.9626 Tf 9.963 0 Td [(H k + j s t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s !# t + s s )]TJ/F10 6.9738 Tf 6.227 0 Td [(i=s )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A : 87

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From.24wehave exp i s H m + p )]TJ/F1 9.9626 Tf 9.962 14.047 Td [( k + j s t + s )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i s )]TJ/F11 9.9626 Tf 9.962 0 Td [(H m )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( k + j s t )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.962 0 Td [(i s !# t + s t i=s )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.133 -8.069 Td [(1+ O =! : Thesameresultsoccurwithcorrespondinglowerbounds.Thuswehavefrom.44that s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 k + j s t + s )]TJ/F10 6.9738 Tf 11.629 3.923 Td [(i s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s + i s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j = )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.133 -8.07 Td [(1+ O =! s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(j : Wehavethusshownthatfor k 2 D A;! s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 X k 2 D A;! i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i + t X j =1 X k 2 D A;! j k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.132 -8.07 Td [(1+ O =! t 2 s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 X k 2 D A;! k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.963 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j : Thissumover j inturnequals,upondivisionby )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 X k 2 D A;! k + j s t + s ks + j m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [( k + j s t + s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(j = O + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 D A;! k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : Wenowdealwiththesecondarytail,thetermswith k 2 U A;! Lemma5.15. Let !> 0 beaconstant.For 1 i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 and 1 j t ,wehaveforlarge p that m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k 2 U A;! k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i = o p=h ; and m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k 2 U A;! k t + s + c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = o p=h : 88

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Proof. Wewillshowtherstresult,asthesecondresultcanbeprovensimilarly.Wehaveby Stirling'sFormulathat .45 k t + s ks = t + s s ks t + s t kt r t + s 2 kst )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A : Thus .46 k t + s + c i ks + b i = t + s s ks + b i t + s t kt + a i r t + s 2 kst )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ O =A : Fortheotherbinomialterm,wehavethat m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i = )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A t + s s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b i t + s t m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .47 1+ h t + s + i )]TJ/F11 9.9626 Tf 9.962 0 Td [(s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b i 1 )]TJ/F11 9.9626 Tf 26.03 6.74 Td [(h t + s + i )]TJ/F11 9.9626 Tf 9.963 0 Td [(s t + s m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.226 0 Td [(a i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Notethatthepowersof t + s =s fromtheresultofLemma5.2,alongwith.46and.47cancel, andalonefactorof t= t + s remainsfromtheotherproportion.Wenowfocusinontheone plus"andoneminus"termsfromtheresultofLemma5.2and.47.Sinceboth )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ =x x is increasingand1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x exp )]TJ/F11 9.9626 Tf 7.748 0 Td [(x ,for k 2 U A;! wehave 1+ h t + s + i )]TJ/F11 9.9626 Tf 9.963 0 Td [(s t + s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i p )]TJ/F10 6.9738 Tf 6.226 0 Td [(ks )]TJ/F10 6.9738 Tf 6.227 0 Td [(b i 1+ 1+ i )]TJ/F11 9.9626 Tf 9.963 0 Td [(s = [ h t + s ] !s !hs exp )]TJ/F11 9.9626 Tf 9.963 0 Td [(a h + i )]TJ/F11 9.9626 Tf 9.962 0 Td [(s t + s ; .48 forsome a 2 ; 1,and .49 1 )]TJ/F11 9.9626 Tf 11.158 6.739 Td [(h + i )]TJ/F11 9.9626 Tf 9.963 0 Td [(s = t + s m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.227 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 exp )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( h + i )]TJ/F11 9.9626 Tf 9.963 0 Td [(s t + s : Thustheproductof.48and.49isboundedaboveby .50exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(ah 1 )]TJ/F11 9.9626 Tf 17.985 6.74 Td [(s )]TJ/F11 9.9626 Tf 9.963 0 Td [(i h t + s exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(bh ; forsome b 2 ; 1.Since1+ x exp x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 = 2forall x 0,wehave .51 1+ h p )]TJ/F10 6.9738 Tf 6.227 0 Td [(p exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(h + h 2 2 p : 89

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Forsucientlysmall x 0wehave1 )]TJ/F11 9.9626 Tf 9.778 0 Td [(x exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F11 9.9626 Tf 9.777 0 Td [(x 2 = 2.As h = o m ,wehaveas p !1 that .52 1 )]TJ/F11 9.9626 Tf 12.662 6.74 Td [(h m )]TJ/F10 6.9738 Tf 6.227 0 Td [(m exp h + h 2 m : Therefore,for p largeenough,wehavefromtheresultof.50,.51,.52that 1+ h p )]TJ/F10 6.9738 Tf 6.227 0 Td [(p 1+ h + i )]TJ/F11 9.9626 Tf 8.855 0 Td [(s = t + s p )]TJ/F11 9.9626 Tf 8.856 0 Td [(ks )]TJ/F11 9.9626 Tf 8.855 0 Td [(b i p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks )]TJ/F10 6.9738 Tf 6.226 0 Td [(b i 1 )]TJ/F11 9.9626 Tf 11.555 6.74 Td [(h m )]TJ/F10 6.9738 Tf 6.227 0 Td [(m 1 )]TJ/F11 9.9626 Tf 10.051 6.74 Td [(h + i )]TJ/F11 9.9626 Tf 8.856 0 Td [(s = t + s m )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt )]TJ/F11 9.9626 Tf 8.855 0 Td [(a i )]TJ/F8 9.9626 Tf 8.856 0 Td [(1 m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .53 exp h )]TJ/F11 9.9626 Tf 7.749 0 Td [(b + h 2 p + h m : Asforthetermsunderthesquarerootsin.46,.47,andtheresultofLemma5.2,wehave s + t kt = O t + s pt and mp m + p pt t + s : Wealsohave m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ As )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i As )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .54 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt=s )]TJ/F8 9.9626 Tf 9.963 0 Td [( As )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i As )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt=s )]TJ/F11 9.9626 Tf 9.963 0 Td [(a i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 =1+ o : Thustheproductofthesquareroots )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(aswellastheextrafactorof t= t + s is O .Since h = o p and h = o m thereexistsaconstant c> 0suchthatforsucientlylarge p via.53: m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b i .55 O )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ O =A exp h )]TJ/F11 9.9626 Tf 7.749 0 Td [(b + h 2 p + h m exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(ch ; wherethelastlinefollowssinceboth h=p and h=m goto0.Summingoverthisrangeof k ,thereare atmost !h terms.Thus m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p=s )]TJ/F10 6.9738 Tf 6.226 0 Td [(A X k = p=s )]TJ/F10 6.9738 Tf 6.226 0 Td [(!h k t + s + b i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i !h exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(ch = !p=h h 2 =p exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(ch = !p=h o exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(ch = o p=h : 90

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Thereareaconstantnumberofeachtypeofterm )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(around s 2 + t 2 = 2ofthem .Therefore,the sameresultholdsfortheentiresums m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 X k 2 U A;! i k t + s + c i ks + b i m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(c i )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.963 0 Td [(b i and m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t X j =1 X k 2 U A;! j k t + s + c )]TJ/F10 6.9738 Tf 6.226 0 Td [(j ks + b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Weshallreplacethesetwosumswiththesum t 2 X k 2 U A;! k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Lemma5.16. Forsucientlylarge p X k 2 U A;! k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k = o p=h : WeproceedsimilarlytoLemma5.12,reducingtosumsappearinginLemma5.15. Wehavethusshownthat G m;p ; s;t = O A + o p=h + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.133 -8.07 Td [(1+ O =! m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = O A + o p=h + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ O =A )]TJ/F8 9.9626 Tf 9.133 -8.07 Td [(1+ O =! m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k =0 m + p +1 k t s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : Letting p !1 ,welet A and beaslargeasneeded,andLemma5.5gives G m;p ; s;t t t + s 2 p ms )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt : Corollary5.17. TheresultofTheorem5.14holdsforarbitrarypositive s and t Proof. AsintheproofofCorollary5.13,if t=s isrational, G m;p ; s;t = nG m;p ; r;q forsome constant n andcoprimepositiveintegers q and r ,and nG m;p ; r;q = n q q + r 2 p mr )]TJ/F11 9.9626 Tf 9.963 0 Td [(pq )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o = t t + s 2 p ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o : Theremainingcasesfollowbycontinuity. 91

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5.5. Urnswith ms=pt > 1 Theremainingcase,forwhich ms=pt > 1as p !1 ,seems tobediculttoanalyze.Since G m;p ; s;t turnsouttobe O forlarge p ,wecannotaordto cutothetailsalaLemma5.8.Foldingthetwodoublesumstogetheralsofails.Withoutadirect connectiontothesum p X k =0 k t + s s k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s s p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k outsideofthecase s =1,wecannotuseresultsdealingwiththetailofabinomialdistribution.So, weareforcedtomanipulateintoaformwith s =1.With s =1,wehavethebinomialformtowork with: G m;p ;1 ;t = )]TJ/F11 9.9626 Tf 9.636 6.74 Td [(t 2 + m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t 2 p X k =0 m + p +1 k t p )]TJ/F10 6.9738 Tf 6.227 0 Td [(k : ThefollowingfollowsimmediatelyfromLemma5.7. Theorem5.18. If m= pt > 1 as p !1 ,then G m;p ;1 ;t t +1 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = t t +1 2 p m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt : Wecanshow G m;p ; s;t isboundedwhen ms= pt > 1usingTheorem5.18. Lemma5.19. Supposethatas p !1 ms=pt > 1 .Then,as p !1 G m;p ; s;t s t +1 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Proof. Forany p ,thereisaminimalinteger^ p suchthat^ ps p .ThenbyLemma2.1,Lemma2.6, andLemma2.9,wehavethat G m;p ; s;t G m; ^ ps ; s;t = sG m; ^ ps ;1 ;t=s sG m; ^ p ;1 ;t : If < 1 ,wehave p ^ ps forlargeenough p .Therefore, ms pt implies m ^ pt aswell.Then,by Theorem5.18, G m; ^ p ;1 ;t t +1 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Wehavethusshownthat G m;p ; s;t isboundedfromabove. Insomecircumstances,wecanusethemonotonicityin s and t toproduceapossiblybetter bound. 92

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Lemma5.20. Supposethatas p !1 ms=pt > 1 .Ifinaddition m p d t=s e 1 > 1 ; thenas p !1 G m;p ; s;t s 2 d t=s e +1 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+ o : Proof. Wehavethat G m;p ; s;t = sG m;p ;1 ;t=s ..6impliesthat G m;p ;1 ;t=s G )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(m;p ;1 ; d t=s e : Since d t=s e isaninteger,underthegivenconditionswecanuseTheorem5.18: G )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m;p ;1 ; d t=s e = 1 2 d t=s e +1 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o ; as p !1 : Multiplyingthroughby s gives G m;p ; s;t s G )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m;p ;1 ; d t=s e s 2 d t=s e +1 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o ; asdesired. Wecanusethesamemethodtoproduceabetterlowerboundaswell. Lemma5.21. Supposethatas p !1 ms=pt > 1 ,andthat t>s .Dene 0 sothat m p b t=s c 0 as p !1 : Then G m;p ; s;t s 2 b t=s c +1 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o : TheprooffollowsthatofLemma5.20,andisomitted.If s>t ,weusezeroasalowerbound. Remark .If p isboundedas m goestoinnitythatis, = 1 ,then G m;p ; s;t 0underthese circumstances,astheprobabilityofplacingabetalsogoestozero. 5.6. Urnswith pt )]TJ/F11 9.9626 Tf 10.402 0 Td [(ms 0 and p p = o pt )]TJ/F11 9.9626 Tf 10.402 0 Td [(ms Fortheremainingurnswith pt )]TJ/F11 9.9626 Tf 10.402 0 Td [(ms 0, observethatif p p = o pt )]TJ/F11 9.9626 Tf 9.352 0 Td [(ms ,then pt )]TJ/F11 9.9626 Tf 9.352 0 Td [(ms ismuchlargerthan G p;m ; t;s ,fromTheorems5.14 and5.20.Therefore,thefollowingisaresultoftheAntiurnTheorem. Theorem5.22. If pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms = p p !1 as p !1 ,then G m;p ; s;t pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms: 93

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6. TheProbabilityofMinimalGain Usinganoptimalstrategyforthe m;p ; s;t urn,theplayerisguaranteedtogainmax f 0 ;pt )]TJ/F11 9.9626 Tf 8.456 0 Td [(ms g Inthischapter,weshallexaminetheprobabilitytheplayergainsmax f 0 ;pt )]TJ/F11 9.9626 Tf 7.886 0 Td [(ms g ,whichisavariation ontheballotproblem. 6.1. ABriefHistoryoftheBallotProblem. Thegeneralizedballotproblemcanbestatedas follows:Inanelection,candidate A has a votes,andcandidate B has b votes,with a b and 0 axedrealnumber.Whatistheprobabilitythat,asthevotesarecounted,thatithenumberof votesfor A isalwaysgreaterthan timesthenumberofvotesfor B ,oriithenumberofvotesfor A isalwaysatleast timesthenumberofvotesfor B ?Anequivalentformulationisasfollows: A has a votesofweightone,and B has b votesofweight ,andwewishtocalculatetheprobability that A alwayshasaigreater,oriigreaterorequal,weightedtotalthan B throughoutthecount. TheproblemwasoriginallyproposedbyM.Bertrand[3]in1887.Heindicatedthatthesolution toiwith =1is a )]TJ/F11 9.9626 Tf 9.971 0 Td [(b = a + b .Thatsameyear,D.Andre[2]gaveadirectproofofBertrand's result.In1924A.Aeppli[1]showedthatthesolutionforinteger 0foriis .1 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b a + b : Thesolutiontoiiforinteger 0followsfromibyaddinganadditionalvotefor A tobe countedrst.Thesolutiontoiiisthus a +1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b a +1 : Forothervaluesof ,thesetwoformulasdonotwork.In1962,Takacs[24]gavethesolutiontoi forarbitrary 0: Theorem6.1. Takacs If a b ,thentheprobabilitythatthenumberofvotesfor A isalways morethan timesthenumberofvotesfor B is a a + b b X j =0 C j b j a + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = a + b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b X j =0 C j a + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(j b )]TJ/F11 9.9626 Tf 9.963 0 Td [(j ; 94

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where C 0 =1 andtheconstants C j ;j =1 ; 2 ;::: aregivenbytherecurrenceformula n X j =0 C j n j b n c + n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 =0 ;n =1 ; 2 ;::: Takacsmadetheimportantobservationthattheseballotnumbersareuniquelydeterminedby thebinomialrecurrencerelation,alongwiththeboundaryconditions.Thegeneralsolutiontoii willbegivenwithTheorem6.10. 6.2. ZeroGainforthe m;p ;1 ;t Urns, t 0 anInteger. When pt )]TJ/F11 9.9626 Tf 10.369 0 Td [(m 0,Theorem3.29 impliesthattheprobabilitytheplayergainszerousinganyoptimalstrategyequals 1 )]TJ/F1 9.9626 Tf 9.962 14.047 Td [( m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q m;p : Theone-to-onecorrespondencespelledoutbyTheorem4.18isactuallytheidentitymapunderthe zero-passstrategy.Thatis,usingthezero-passstrategy,theplayerwillgainzeroifandonlyif X n 0forall n andthus,theplayerwillneverbet,orequivalently, N + =0.Thustheprobability ofzerogainequals m )]TJ/F17 7.9701 Tf 6.586 0 Td [(pt +1 m +1 ,viaCorollary3.14.Thisisversioniioftheballotproblem.Wehave discussedversioniaswell-a+ b )]TJ/F11 9.9626 Tf 9.962 0 Td [(a "tripis,infact,aballotpermutationoftypei. When pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m> 0,theminimumgainis pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m andtheprobabilityofminimumgainequals .21 )]TJ/F1 9.9626 Tf 9.963 14.047 Td [( m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q m;p pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m +1 ; andbyTheorem3.26, .3 Q m;p pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m +1= b m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =t c X k =0 1 kt + k +1 kt + k +1 k m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : Recallthedenitionof N ,thenumberoftimesthattheurnisneutral: N = f n : n 6 = m + p;X n =0 g : Wenowexaminethedistributionof N overthezero-gainrealizationswith m pt Lemma6.2. Suppose m pt ,with t apositiveinteger.Thenusinganyoptimalstrategy, P Playergainszeroand N = k = m )]TJ/F8 9.9626 Tf 9.962 0 Td [( p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : 95

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and P Playergainszeroand N k = m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + kt +1 m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k +1 m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k +1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : Remark .Therstresultwith k =0is.1. Proof. TherstresultfollowsfromTheorem3.16.Sincewehavetheaddedrestriction N = k ,we takeonlythe n = k termfromthesum.So,weconcentrateonthesecondresult. Withthezero-passstrategy,if isazero-gainrealization,then N = N )]TJ/F8 9.9626 Tf 6.725 -3.615 Td [(.Weremovethe+ t "s causingthethenal k events A t .Thisforms k + t "trips.Theinitialportion,uponanaddition ofa )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balltotheend,formsa+ m )]TJ/F11 9.9626 Tf 10.002 0 Td [(pt +1"trip.Therefore,arealizationgainingzerowith N k isinaone-to-onecorrespondencewith+ m +1 )]TJ/F11 9.9626 Tf 10.007 0 Td [(pt )]TJ/F11 9.9626 Tf 10.007 0 Td [(kt "tripscontaining k fewer+ t "s andonemore )]TJ/F8 9.9626 Tf 7.749 0 Td [(1."Thetotalnumberofthelatteris m )]TJ/F11 9.9626 Tf 9.963 0 Td [(pt + kt +1 m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k +1 m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k +1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Wecompletebydividingby )]TJ/F17 7.9701 Tf 5.48 -4.378 Td [(m + p p Remark .Lemma6.2givesacombinatorialproofofCorollaries1iiandiiiof[15]. 6.3. GeneralizedBallotNumbersandZeroGainNumbers. Nowwetakealookat s;t ballotnumbersand s;t -zero-gainnumbers,theirsimilarities,andtheinterplaybetweenthetwo collectionsofnumbers.Webeginwiththe s;t -ballotnumbers. Denition6.3. The s;t -ballotnumber s;t a;b isthenumberofwaysthat,inanelection,a candidatereceiving a votesofweight s remainsaheadinweightoveracandidatereceiving b votes ofweight t throughoutthecountingofthevotes. Remark .Weshalltaketheconventionthat s;t ; 0=1. Clearly,if bt as ,wehave s;t a;b =0,ascandidate B willwin.The s;t -ballotnumbers satisfytherecurrencemostoftenassociatedwiththebinomialcoecients,byconsideringthelast votecounted: s;t a;b = s;t a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;b + s;t a;b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt> 0 orequivalently,for a> 0, s;t a;b = b X k =0 s;t a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;k : 96

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Some ; 3-ballotnumbersaregiveninTable6.1. The s;t -zero-gainnumbersaredenedsimilarly. Denition6.4. The s;t -zero-gainnumber s;t a;b isthenumberofrealizationsfromthe a;b ; s;t urnthatgainzerowhenthezero-passstrategyisused. Remark .Again,weset s;t ; 0=1byconvention. If bt>as ,wehave s;t a;b =0,astheplayerwilldenitelygainatleast bt )]TJ/F11 9.9626 Tf 10.068 0 Td [(as> 0.Wehave thesamebinomialrecurrence: s;t a;b = s;t a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;b + s;t a;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;as )]TJ/F11 9.9626 Tf 9.962 0 Td [(bt 0 ; and s;t a;b = b X k =0 s;t a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;k ;a> 0 : Some ; 3-zero-gainnumbersaregiveninTable6.2. Table6.1. ; 3-ballotnumbersfor0 a 11,0 b 7. a 01234567891011 2 ; 3 a; 0111111111111 2 ; 3 a; 112345678910 2 ; 3 a; 2 0 37121825334252 2 ; 3 a; 3719376295137189 2 ; 3 a; 4 0 3799194331520 2 ; 3 a; 5992936241144 2 ; 3 a; 6 0 6241768 2 ; 3 a; 71768 Lemma6.5. Fornonnegative s t ,and r> 0 ,wehave s;t a;b = rs;rt a;b and s;t a;b = rs;rt a;b : Thisisclearbecause as )]TJ/F11 9.9626 Tf 10.355 0 Td [(bt and ars )]TJ/F11 9.9626 Tf 10.356 0 Td [(brt = r as )]TJ/F11 9.9626 Tf 10.356 0 Td [(bt willalwayshavethesamesign.Thus s;t a;b isrelatedtoproblemiand s;t a;b isrelatedtoproblemii,with = t=s .Forthe s;t -zero-gainnumbers,theboundaryisincludedintermsoftheballotproblem,tiesbetween A and B areallowed,whiletheboundaryisnotincludedforthe s;t -ballotnumbersintermsofthe zero-gainproblem,theweightoftheurnstaysnegativewhiletheurnisnonempty.Therefore,we 97

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have s;t a;b s;t a;b ; forall a b s t; sinceeachballotpermutationcorrespondswithazero-gainrealization.If t=s isirrational,problems iandiiareequivalent,astherecannotbeanytiesoncethevotecountstarts.Thatis, as )]TJ/F11 9.9626 Tf 8.439 0 Td [(bt =0 ifandonlyif a = b =0.Therefore,wehave s;t a;b = s;t a;b ; forall a and b ,if t=s isirrational. Table6.2. ; 3-zero-gainnumbersfor0 a 11,0 b 7. a 01234567891011 2 ; 3 a; 0111111111111 2 ; 3 a; 112345678910 2 ; 3 a; 2 2 59142027354454 2 ; 3 a; 39234370105149203 2 ; 3 a; 4 23 66136241390593 2 ; 3 a; 51363777671360 2 ; 3 a; 6 377 11442504 2 ; 3 a; 72504 Otherresultsincludethefollowing. Lemma6.6. Suppose as = bt .Then s;t a;b = t;s b;a ThisisadirectconsequenceoftheAntiurnTheoremandtheReversalLemma. Lemma6.7. Suppose as )]TJ/F11 9.9626 Tf 10.375 0 Td [(bt 0 ,and s and t arepositiveintegerswith gcd s;t =1 : Thenwe have s;t a;b )]TJ/F11 9.9626 Tf 9.963 0 Td [( s;t a;b = b b=s c X k =1 s;t a )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt;b )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks s;t kt;ks : Proof. Foreachzero-gainrealization,thereisarsttimetheurnisneutral,since X m + p =0.From theballotperspective,thereisalwaysalasttie.Asgcd s;t =1,theurncanonlybeneutral whenthereare kt )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballsand ks + t "ballsleftfor0 k b b=s c .Theballsleftformavalid s;t -zero-gainrealizationwith kt )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "ballsand ks + t "balls,whiletheballsthathavebeendrawn formavalid s;t -ballotpermutationontheremaining a )]TJ/F11 9.9626 Tf 10.22 0 Td [(kt votesfor A and b )]TJ/F11 9.9626 Tf 10.22 0 Td [(ks votesfor B withthetwoeventsindependent. 98

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Ifgcd s;t > 1,werefertotheresultobtainedafterusingLemma6.5.If as = bt ,thentheonly contributingtermis k = b=s )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(whichisanintegersincegcd s;t =1 .If t=s isirrational,then theonlycontributingtermis k =0,makingthe s;t -ballotand s;t -zero-gainnumbersequal,as alreadynoted. Aseconddecompositionof s;t a;b canbemadeaswell: Lemma6.8. Suppose as )]TJ/F11 9.9626 Tf 10.375 0 Td [(bt 0 ,and s and t arepositiveintegerswith gcd s;t =1 : Thenwe have s;t a;b )]TJ/F11 9.9626 Tf 9.963 0 Td [( s;t a;b = b b=s c X k =1 s;t a )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt;b )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks s;t kt;ks )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Proof. Each a;b ; s;t -zero-gainnumberthatisnota a;b ; s;t -ballotnumberhas,intermsofthe ballotproblem,arsttieafterthevotecountstarts.Thevoteforcingthattieisfor B .Ifthe rsttieoccurswhen k t + s voteshavebeencounted,thentherst k t + s )]TJ/F8 9.9626 Tf 10.74 0 Td [(1votesforma kt;ks )]TJ/F8 9.9626 Tf 10.792 0 Td [(1; s;t -ballotnumber,withtheremainingvotesmakinga a )]TJ/F11 9.9626 Tf 10.791 0 Td [(kt;b )]TJ/F11 9.9626 Tf 10.791 0 Td [(ks ; s;t -zero-gain number. Adecompositionof s;t a;b isnotsoeasy.When s =1,wehavethescenariothatbridgesthe gapbetweeniandii. Lemma6.9. 1 ;t a;b = 1 ;t a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;b forany a> 0 andpositiveinteger t Generally,wehave s;t a;b = 1 ; a;b ,with = t=s and 1 ; a;b )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(a + b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 givenbytheformula inTheorem6.1.Theconstants C 1 ;C 2 ;::: aredeterminedbytheboundaryconditions 1 ; )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [(b b c ;b =0 : Forthezero-gainnumbers,wecanobtainthemsimilarlybyadjustingtheboundaryconditions.We haveinthiscase 1 ; )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [(b b c ;b =0 ; if b isnotaninteger. If b isaninteger,thenwehaveatie,butthistimetiesarepermitted.Thus,wetakeavoteaway from A tollouttheboundarycondition: 1 ; b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;b =0 ; if b isaninteger. 99

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If isirrational,wewillnothaveanyties,sowereducebacktotheballotnumbers.Otherwise,we havethefollowing. Theorem6.10. Suppose = t=s ,with s and t positiveintegers.If a b ,then a + b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 ; a;b = a + b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b X j =0 D j a + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(j b )]TJ/F11 9.9626 Tf 9.963 0 Td [(j ; where D 0 =1 andtheconstants D j ;j =1 ; 2 ;::: aregivenbytherecurrenceformula n X j =0 D j b n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 =s c + n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(j n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j =0 ;n =1 ; 2 ;::: Proof. Since = t=s n hastheform x=s ,with x aninteger,and n )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 s = 8 > > < > > : b n c ; if n isnotaninteger, n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; if n isaninteger. Thus,wehavetheproperboundaryconditions. Example .With =1,wehave D j =1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(j for j =0 ; 1 ;::: andwehave 1 ; 1 a;b = b X j =0 )]TJ/F11 9.9626 Tf 9.962 0 Td [(j a + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(j b )]TJ/F11 9.9626 Tf 9.962 0 Td [(j = a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b +1 a +1 a + b a ; asexpected. Weconcludewithsomespecialcases. Proposition6.11. Suppose s and t arepositiveintegerswith as )]TJ/F11 9.9626 Tf 9.962 0 Td [(bt =1 .Then s;t a;b = 1 a + b a + b a : ThisLemmaisadirectcorollaryofatheorembyRaney[16]: Theorem6.12. Raney If h x 1 ;x 2 ;:::;x m i isanysequenceofintegerswhosesumis +1 ,exactly oneofthecyclicshifts h x 1 ;x 2 ;:::;x m i ; h x 2 ;x 3 ;:::;x 1 i ;:::; h x m ;x 1 ;:::;x m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i hasallofitspartialsumspositive. 100

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Here,thesequenceofintegersisoflength a + b ,andiscomposedentirelyof+ s "sand )]TJ/F11 9.9626 Tf 7.749 0 Td [(t "s, afterwhichwemultiplythroughby )]TJ/F8 9.9626 Tf 7.749 0 Td [(1toobtaincorrespondingrealizations.Anoutlineoftheproof canbefoundin[10,pp.359-360]. Figure6.1. Azero-passzero-gainrealizationfromthe ; 4;2 ; 3urnandits corresponding ; 7;3 ; 2ballotpermutationreectedthroughthe n -axisandshifted leftoneunit. Wehaveacorollarysince s;t a;b = s;t a;b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1if as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt =1. Corollary6.13. If s and t arepositiveintegerswith as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt = t +1 ,then s;t a;b = 1 a + b +1 a + b +1 a = 1 b +1 a + b a : Proposition6.11alsohasaconsequencewithsome s;t -zero-gainnumbers. Proposition6.14. Suppose s and t areintegerswith as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt = t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .Then s;t a;b = 1 a + b +1 a + b +1 a = 1 b +1 a + b a : Proof. Startwithazero-gainrealization fromthe a;b ; s;t urn.Thisrealizationisnaturally associatedwiththerealization^ fromthe b;a ; t;s antiurnbychangingthesignsontheballs.We nowattacha )]TJ/F11 9.9626 Tf 7.748 0 Td [(t "balltothe beginning of^ ,formingarealization fromthe b +1 ;a ; t;s urn, 101

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with X n = X n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ^ 0for1 n a + b ,and X 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1.Anexamplewithazero-gain realizationfromthe ; 4;2 ; 3urnisgivenwithFigure6.1.Thenthereversedrealization R satises X n R )]TJ/F8 9.9626 Tf 18.265 0 Td [(1 < 0for nm equals m + p p )]TJ/F13 6.9738 Tf 9.963 13.311 Td [(b m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =t c X k =0 1 kt + k +1 kt + k +1 k m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Whenweapplytheantiurnmaptotheserealizations,theybecomezero-gainrealizationsviaa dierentstrategyfromthe p;m ; t; 1urn,andthusthe p;m ;1 ; 1 =t urn.Therefore,wehave: Theorem6.15. Suppose t isapositiveinteger.Then,if a b=t wehave .4 1 ; 1 =t a;b = a + b b )]TJ/F13 6.9738 Tf 9.963 13.311 Td [(b b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =t c X k =0 1 kt + k +1 kt + k +1 k a + b )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(k : Asimilarprocedurecannetthe ; 1 =t -ballotnumbers. Theorem6.16. Suppose t isapositiveinteger.Then,if a b=t wehave .5 1 ; 1 =t a;b = a + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 b )]TJ/F13 6.9738 Tf 9.963 13.311 Td [(b b=t c)]TJ/F7 6.9738 Tf 9.879 0 Td [(1 X k =0 t kt + k + t kt + k +1 k a + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(t a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : Proof. Observethatarealizationfromthe b;a ;1 ;t urnwith at>b and X n > 0for n 6 = a + b is, infact,a t; 1-ballot )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( ; 1 =t -ballot permutationupontheantiurnreection.Wecalculatetheir number.Obviously,thelastballmustbea+ t ,"soweremoveit.Uponreversingtheremaining part,weseethatthisisarealizationfromthe b;a )]TJ/F8 9.9626 Tf 10.291 0 Td [(1;1 ;t urnwith X n
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donewiththetraditionalballotproblemLemma6.9.Withsomemanipulation,wecanarriveat thefollowingalternativeformsfor 1 ; 1 =t a;b and 1 ; 1 =t a;b : 1 ; 1 =t a;b = a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t )]TJ/F10 6.9738 Tf 6.227 0 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 t +1 X j =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 j t a )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(j t +1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(j at )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(jt j ; 1 ; 1 =t a;b = at )]TJ/F10 6.9738 Tf 6.227 0 Td [(b )]TJ/F10 6.9738 Tf 6.226 0 Td [(t t +1 X j =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 j a )]TJ/F11 9.9626 Tf 9.962 0 Td [(j t +1+1 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(j t +1+1 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(j at )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F11 9.9626 Tf 9.963 0 Td [(jt j : Theapplicationofthegeneralizedballotnumberstotheproblemofminimalgainissummarized withthenexttheorem. Theorem6.17. Forthe m;p ; s;t urn: 1 If pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms 0 ,theprobabilitytheplayerwillgainzerousinganyoptimalstrategyis m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 ;t=s m;p ; where 1 ;t=s m;p isgivenbytheresultofTheorem6.10if t=s isrational,and 1 ;t=s m;p = 1 ;t=s m;p isgivenbytheresultofTheorem6.1if t=s isirrational. 2 If pt )]TJ/F11 9.9626 Tf 10.183 0 Td [(ms 0 ,thentheprobabilitytheplayergains pt )]TJ/F11 9.9626 Tf 10.184 0 Td [(ms usinganyoptimalstrategyis thesameastheprobabilityofzerogainforthe p;m ; t;s urn. Proof. If pt )]TJ/F11 9.9626 Tf 10.771 0 Td [(ms 0,iftheplayerusesthezero-betstrategy,thenzero-gainrealizationsfrom the p;m ; t;s urnwiththezero-passstrategycorrespondtotheminimalgainrealizationsfromthe m;p ; s;t urnviathenaturalantiurnmap. 6.4. AnotherExtensionoftheBallotProblem. Wehavenotedthatthesolutionsthusfarto thetwoballotproblemsarenotprettyoutsideofafewnicecases.Now,wewillextendtheproblem again,butthistimewewillexpanduponthe answer instead.Recallthatwhen isapositive integer,thesolutionstotheballotproblem,versionsiandii,arerespectively .6 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b a + b a + b a ; a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b +1 a +1 a + b a : Wewillpresentaquestionwithananswerfor rational,reducingto.6when isapositive integer.Todothat,weshallneedmorenotation. 103

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Denoteby i s;t a;b asthenumberofballotpermutationsforwhichcandidate A nevertrailsby morethan i inweightoncethevotecountstarts.Thus, s;t a;b isthe s;t -zero-gainnumberand )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 s;t a;b isthe s;t -ballotnumber.Wenowpresenttheresult. Theorem6.18. Suppose s and t arepositiveintegerswith gcd s;t =1 .If as bt ,then .7 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 i s;t a;b = s a +1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(bt a +1 a + b b : If as>bt ,then .8 s X i =1 )]TJ/F10 6.9738 Tf 6.226 0 Td [(i s;t a;b = as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt a + b a + b b : Observethatwehaveareductionto.6when s =1.Theorem6.18isactuallyaspecialcaseof amoregeneralresultfoundbyIrvingandRattan[12].Weshallpresenttheirresultshortly, butweshallneedtoadopttheirnotation. Aparticlebeginsatthepoint ; 0,andateachstagemoveseitheroneunittotherightorone unitup,eventuallyreachingthepoint n;m Denition6.19. A weak m -partcompositionof n isalist f = f 0 ;:::;f m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 with f j anonnegative integerforeach j and P m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 i =0 f j = n Each m -partcompositionof n inducesthepiecewiselinearboundary @ f denedby x = f i y )]TJ/F11 9.9626 Tf 9.963 0 Td [(i + i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 f j ;y 2 [ i;i +1] : Anypointorpathlyingweaklybelowtheboundary @ f issaidtobe dominated by f .Irvingand Rattan'sresultrelatestothecyclicshiftsof f .The j th cyclicshiftof f is f h j i = f m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(j ;:::;f m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ;f 0 ;:::;f m )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(j : Forany f ,denoteby D f asthenumberofpathsdominatedby @ f .Themainresultof[12]isthe following: Theorem6.20. IrvingandRattan Let f beaweak m -partcompositionof n andlet T = a;b where 0 a n and 0 b m .Ifthepoint T 0 = a +1 ;b liesweaklytotherightof @ f h j i forall 104

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j ,then m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =0 D )]TJ/F42 9.9626 Tf 4.566 -8.07 Td [(f h j i = m a +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(nb a +1 a + b b : Forthe m;p ; s;t scenario,ourboundaryistheline y = sx=t andtheurnemptiesat ; 0so thatrealizationsruninreverse,andouraimistoreplace y = sx=t withaboundarymoreinline withIrvingandRattan'sresult.Notethatweonlyneedtogureoutwhatoccursuptothepoint t;s ,asthepatternresetsatthatpoint.Lemma6.21givesthecompositionofthefundamental periodofthepiecewiselinearboundaryweshallusetomimetheline y = sx=t Lemma6.21. Suppose gcd s;t =1 ,and r istheintegersatisfying r
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Figure6.2. For s =8, t =3,wehave r =0, f = ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0,andthe piecewiseboundary @ f Lemma6.22. Let f beasdenedinLemma6.21.If kt )]TJ/F1 9.9626 Tf 9.069 7.472 Td [(P k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 sf i = )]TJ/F11 9.9626 Tf 7.749 0 Td [(j ,thenthepathsdominated by @ f h s )]TJ/F10 6.9738 Tf 6.227 0 Td [(k i satisfy X n j forall n Proof. If kt )]TJ/F1 9.9626 Tf 9.963 7.472 Td [(P k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 sf i = )]TJ/F11 9.9626 Tf 7.749 0 Td [(j ,thenfor k<` s 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(s `t )]TJ/F10 6.9738 Tf 10.391 12.453 Td [(` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 sf i 0.9 impliesthat 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(s + j ` )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t )]TJ/F10 6.9738 Tf 10.392 12.453 Td [(` )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i = k sf i j: .10 Since s )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t )]TJ/F10 6.9738 Tf 10.179 12.453 Td [(s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i = k sf i = j and.9holdsfor ` k ,wehavethat.10holdsfor k<` s )]TJ/F8 9.9626 Tf 10.133 0 Td [(1+ k where f i isunderstood tobe f i mod s .Therefore,wehavethat ` )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t )]TJ/F10 6.9738 Tf 10.391 12.453 Td [(` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i = k sf i s + k ` = k +1 = f 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(s + j;:::;j g : Thisshowsthattheboundary @ f h s )]TJ/F10 6.9738 Tf 6.227 0 Td [(k i dominatesanypathsatisfying X n j forall n Toextendthistothe m;p ; s;t urns,wemerelyrepeattheboundary @ f asmanytimesas necessarytocovertheinitialurnpoint p;m ,inwhichcase m timeswillcertainlydothejob.Let f m denotethegeneralizedboundary )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(theterminusof @ f m is ms;mt .TheresultofTheorem6.20 106

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isthatfor as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt 0, .11 ms )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 D @ )]TJ/F42 9.9626 Tf 4.566 -8.07 Td [(f m h j i = m )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(s a +1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(bt a +1 a + b a : Wethenobservethatonly s oftheboundaryshiftsareuniqueduetotherepetitionof f ,witheach uniqueshiftrepeated m times.Thus.11impliesthat .12 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j =0 D @ )]TJ/F42 9.9626 Tf 4.566 -8.07 Td [(f m h j i = s a +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt a +1 a + b a : TheresultofLemma6.22holdswith f m .Asaresult,foreach0 j s )]TJ/F8 9.9626 Tf 10.507 0 Td [(1thereisaunique 0 i s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1with D f m h j i = i s;t a;b : TherstresultofTheorem6.10nowfollows.Adjoininga )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "totheend )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(inthelatticepath version,startingfromtheinitialpoint ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1instead ,givesthesecondresult.Thatis, s X i =1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(i s;t a;b = s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 i s;t a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;b = as )]TJ/F11 9.9626 Tf 9.963 0 Td [(bt a a + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = as )]TJ/F11 9.9626 Tf 9.962 0 Td [(bt a + b a + b a : ThiscompletestheproofofTheorem6.10.Observethatsince i s;t a;b isincreasingin i ,wecanuse theresultsofTheorem6.10toobtainupperandlowerboundsonthegeneralizedballotnumbers. 6.5. TheFirstCrossing. Recallthatwhen t isapositiveintegerand m
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Proposition6.24. Suppose s and t arepositiveintegers,and ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt> 0 .Then P X =0= m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b p=s c X k =0 k t + s ks s;t m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt;p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks : Proof. Givenarealization with X =0,wewillhave = m + p )]TJ/F11 9.9626 Tf 10.173 0 Td [(k t + s with0 k b p=s c Therst ballsmustformavalid m )]TJ/F11 9.9626 Tf 10.149 0 Td [(kt;p )]TJ/F11 9.9626 Tf 10.149 0 Td [(ks ; s;t -ballotnumber.Thelast k t + s ballscan bedrawnoutinanyfashion. Proposition6.25. Suppose s and t arepositiveintegers,and ms )]TJ/F11 9.9626 Tf 9.962 0 Td [(pt> 0 .Then P X = s )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 M s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 k t + s + c s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ks + b s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 s;t )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt )]TJ/F11 9.9626 Tf 8.855 0 Td [(a s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F8 9.9626 Tf 8.855 0 Td [(1 ;p )]TJ/F11 9.9626 Tf 8.856 0 Td [(ks )]TJ/F11 9.9626 Tf 8.855 0 Td [(b s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; where M s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 isdenedaccordingtotheconditionssetin .4 Proof. Let bearealizationwith X = s )]TJ/F8 9.9626 Tf 8.553 0 Td [(1. canbedividedintothreeparts:Therst )]TJ/F8 9.9626 Tf 8.553 0 Td [(1balls, forwhich X j X )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1for j< ,the th ball,a )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "ball,andtheremaining m + p )]TJ/F11 9.9626 Tf 9.994 0 Td [( balls. Therst ballsforma s;t -zero-gainrealization,uponremovaloftheremaining m + p )]TJ/F11 9.9626 Tf 8.848 0 Td [( )]TJ/F8 9.9626 Tf 8.847 0 Td [(1balls. Thethirdpartiscomposedof ks + b s )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + t "ballsand kt + a s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "ballswith0 k M s )]TJ/F8 9.9626 Tf 9.196 0 Td [(1, andtheycanbedrawninanyorder.Thus,therst ballsconsistof m )]TJ/F11 9.9626 Tf 9.829 0 Td [(kt )]TJ/F11 9.9626 Tf 9.829 0 Td [(a s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F8 9.9626 Tf 9.829 0 Td [(1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(s "balls sinceanotherisusedforthesecondpart,therstcrossingand p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks )]TJ/F11 9.9626 Tf 9.962 0 Td [(b s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + t "balls. Theremainingprobabilitiescanbecalculated,butbecomeextremelycomplicated,withmore andmorecasestoconsider.Weomitthedetails.UsingLemma2.1,wecanextendtheprevious resultstocovertheurnswith t=s rational.When t=s isirrational,thensincenotwoup-crossings areassociatedwiththesamegain,calculationofthevariousprobabilitiesbecomesverydicultin general. 108

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7. ABayesianApproach Inapplicationofthismodel,thetotalnumberofballsisusuallyknown,thenumberof+ t ballsisnot,andisrandom.Thus,asinChen[7]andBoyce[4],aBayesianapproachwouldbe appropriate.Werandomizetheurnbyassigningeachurnwith n ballsaprobabilityofoccurring. Wemayinterpretthisinitialpriordistribution astheplayer'soutlookoverthe n drawsthatwill follow.Anoptimalstrategycanbedeterminedbasedonthisinitialoutlook ,andinmostcases itwillbequitecomplicated.Thus,ndingaformulafortheexpectedgainwillbeverydicultin general,andwewillfocusmostofoureortsontheoriginalBayesianurnwith s = t =1. WewillstartwiththegeneralBayesianurn,anddeveloptheindicator.3thatdetermines whetherabetshouldbeplacedonthenextballbasedontheoutcomeofthepreviousdraws.Then, workingwiththeoriginalrandomurn,wewillndtwofamiliesofdistributionswitharelatively simplebettingrule.Wewillthencalculatetheexpectedgainforthosedistributions.Forthe remainingdistributions,wepresentanalgorithmsimilartothatofBoyce[4]thatproducesthe expectedgain.Wethenreturnbacktothegeneralrandomurn,takingacloserlookatthecases where isbinomialanduniform,ndinganoptimalbettingstrategyandcalculatingtheexpected gainforboth.Finally,weadaptthealgorithmsothatitproducestheexpectedgainforany s t and Remark .Inthischapter,thenumberof+ t "balls p isnotknown,andisarandomvariable. 7.1. Preliminaries. Webeginbydeningtherandomacceptanceurn. Denition7.1. Therandomacceptanceurnwith n ballsofvalue )]TJ/F11 9.9626 Tf 7.749 0 Td [(s and+ t andinitialprior distribution = f q j g n j =0 ofthenumberof+ t "ballsintheurniscalledthe n; ; s;t urn Weseekanoptimalstrategymaximizingtheexpectedgain,whichweshalldenoteas G n; ; s;t Asin[7],Let n = m + p bethetotalnumberofballsintheurn,andlet betheinitialprior distributionoftherandomvariable p ,andlet Y i denotetheweightofthe i th balldrawn.Let A n; ; s;t betheexpectedgainifabetisplacedontherstball,andanoptimalBayesianbetting policyisfollowedthereafter;Let B n; ; s;t betheexpectedgainifnobetisplaced,andanoptimal 109

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Bayesianbettingpolicyisfollowedthereafter.Let G n; ; s;t =max f A n; ; s;t ; B n; ; s;t g denotethevalueoftheurnwith n ballsandpriordistribution Let y 1 betheweightoftherstdrawnball.Then A n; ; s;t = Z y 1 + G )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ; y 1 ; s;t dy 1 ; B n; ; s;t = Z G )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ; y 1 ; s;t dy 1 ; where y 1 istheposteriordistributionofthenumberofballsofweight+ t "aftertherstdraw giventhat Y 1 = y 1 .Then A n; ; s;t B n; ; s;t ifandonlyif Z y 1 dy 1 = t Y 1 = t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s Y 1 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(s 0 ; wewouldbetif t Y 1 = t s Y 1 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(s ,i.e. Y 1 = t s= s + t .Therefore,anoptimal Bayesianbettingpolicycanbestatedasfollows:For0 k n )]TJ/F8 9.9626 Tf 10.249 0 Td [(1,betonthe k +1 th drawif andonlyif t Y k +1 = t j y 1 ;y 2 ;:::;y k s Y k +1 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(s j y 1 ;y 2 ;:::;y k ; i.e. Y k +1 = t j y 1 ;y 2 ;:::;y k s= s + t ,where j y 1 ;y 2 ;:::;y k istheposteriordistributionof thenumberof+ t "ballsgiventhat Y 1 = y 1 Y 2 = y 2 ,..., Y k = y k Remark .If Y k +1 = t j y 1 ;y 2 ;:::;y k = s= s + t ,thentheexpectedgainonthenextdrawnball iszero.Hence,thepolicyofbettingifandonlyif Y k +1 = t j y 1 ;y 2 ;:::;y k >s= s + t isalso optimal.Sincethisstrategygivesthesameexpectedgain,weshallnotusethisstrategymuchin thischapter.Weshallusethisstrategywhenwediscusstheruinproblem,inChapter8. Sinceeachnonrandomacceptanceurnhasanantiurn,theBayesianversionpossessesitsown versionoftheantiurnproperty. Theorem7.2. Let = f q j g n j =0 ,where q j istheprobabilitytheurncontains j + t "ballsinitially, andlet = P n j =0 jq j .Dene R = f w j g n j =0 sothat w j = q n )]TJ/F10 6.9738 Tf 6.226 0 Td [(j forall j .Then .1 G n; ; s;t )]TJETq1 0 0 1 290.284 159.334 cm[]0 d 0 J 0.398 w 0 0 m 7.833 0 l SQBT/F11 9.9626 Tf 290.284 151.132 Td [(G n; R ; t;s = t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(ns: Proof. WeuseasimilarstrategytotheproofoftheAntiurnTheorem.AdamandBettywillplay the n; ; s;t urnsimultaneously.Adamwillusetheprimaryoptimalstrategy,whileBettywill 110

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betifandonlyifAdamdoesnotbet.Then,exactlyoneofthetwowillbetoneachballdrawnfrom theurn.Therefore,theircombinedexpectedgainwillbe t + s )]TJ/F11 9.9626 Tf 10.316 0 Td [(ns .Adam'sexpectedgainis G n; ; s;t ,thusitremainstoshowthatBetty'sexpectedgainis )]TJETq1 0 0 1 404.745 666.426 cm[]0 d 0 J 0.398 w 0 0 m 7.833 0 l SQBT/F11 9.9626 Tf 404.745 658.223 Td [(G n; R ; t;s Bettyplacesabetonthe k +1 th ballifandonlyif Y k +1 = t j y 1 ;:::;y k < s t + s : SinceshebetsifandonlyifAdampasses,herexpectedgainisminimal.Therefore,ifalloftheball weightsweremultipliedby )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,shewouldbemaximizingherexpectedgain.Upondoingthis,we ndthat R istheinitialdistributionofthisantiurn.Letting Y 0 i = )]TJ/F11 9.9626 Tf 7.749 0 Td [(Y i and y 0 i = )]TJ/F11 9.9626 Tf 7.749 0 Td [(y i ,Bettybets ifandonlyif R Y 0 k +1 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(t j y 0 1 ;:::;y 0 k < s t + s R Y 0 k +1 = s j y 0 1 ;:::;y 0 k > t t + s : Thisisanoptimalstrategyforthe n; R ; t;s urn.Therefore,Betty'sexpectedgainis )]TJETq1 0 0 1 482.836 449.093 cm[]0 d 0 J 0.398 w 0 0 m 7.833 0 l SQBT/F11 9.9626 Tf 482.836 440.891 Td [(G n; R ; t;s and.1isshown. Remark .If issymmetric,thatis, = R ,then = n= 2andTheorem7.2impliesthat .2 G n; ; s;t )]TJETq1 0 0 1 267.444 368.606 cm[]0 d 0 J 0.398 w 0 0 m 7.833 0 l SQBT/F11 9.9626 Tf 267.444 360.404 Td [(G n; ; t;s = t + s n 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(ns = n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s 2 : 7.2. DeterminingtheBettingRule. Givenaninitialpriordistribution ,whenis Y k +1 = t j y 1 ;y 2 ;:::;y k s t + s ? Let = f q j g n i =0 bethedistributionoftheurnswith n balls,i.e. q j = urncontains j + t "ballsinitially ; 0 j n: Suppose k ballshavebeendrawn,with ` ofthem+ t "balls.For0 j n ,let q j k;` = q j n j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` : Foreach p q p k;` isthenewweight"associatedwiththeurninitiallycontaining p + t "balls, asonly )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(n )]TJ/F17 7.9701 Tf 6.587 0 Td [(k p )]TJ/F17 7.9701 Tf 6.586 0 Td [(` ofthe )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(n p realizationsarestillpossible.Giventhattheurnhas p + t "balls,each realizationmustoccurwithequalprobability,thus q p isdecimatedbythisproportion.Notethat 111

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if `>p thentheweightiszero.Thustheadjustedprobabilitiesare P urnhas p + t "balls j ` oftherst k ballsdrawnwere+ t "= q p k;` n X j =0 q j k;` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Giventhattheurnhas p + t "balls,theprobabilitythenextballisa+ t "equals p )]TJ/F11 9.9626 Tf 9.991 0 Td [(` = n )]TJ/F11 9.9626 Tf 9.991 0 Td [(k Thus,weshouldplaceabetonthenextballif n X i =0 i )]TJ/F11 9.9626 Tf 9.963 0 Td [(l n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k q i k;l n X j =0 q j k;l s t + s : Wehavethefollowingresult,usingadierentformofthisequation. Theorem7.3. Forthe n; ; s;t urn,if k ballshavebeendrawn,with ` ofthem + t ,"thenan optimalacceptancepolicyistoacceptthe k +1 th ballifandonlyif .3 n X j =0 q j k;` j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(s n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k 0 : Remark .Iftheplayerbetsifandonlyif n X j =0 q j k;` j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(s n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k > 0 ; thenhisstrategyisalsooptimal,sincewhenequalityholdsin.3,theexpectedweightofthenext ballequalszero. 7.3. TheOriginalRandomUrn. Using.3,wendanexpandedsetofinitialdistributionsfor whichthepolicystatedin[7]isoptimal.Asecondfamilyofdistributions,forwhichtheopposite ruleisoptimal,willalsobediscussed. Lemma7.4. Supposetheprobabilitydistribution = f q j g n j =0 satises q j = q n )]TJ/F10 6.9738 Tf 6.226 0 Td [(j .Thenanoptimal bettingstrategyforthe n; ;1 ; 1 urnis .4 betonthe k +1 th ballifandonlyif k X i =1 Y i 0 ifandonlyif 0 j 1 j 2 n= 2 implies .5 n j 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q j 1 n j 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q j 2 : 112

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Proof. Supposethatthestrategyindicatedby.4isoptimalforthe n; ;1 ; 1urn.If n )]TJ/F8 9.9626 Tf 9.988 0 Td [(1balls havebeendrawn, ` ofthem+1,"thentheindicator.3reducesto .6 ` +1 X j = ` n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q j 2 j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = n ` +1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q ` +1 )]TJ/F1 9.9626 Tf 9.962 14.047 Td [( n ` )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 q ` : If2 `
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Since p n )]TJ/F11 9.9626 Tf 10.047 0 Td [(k +2 ` = 2,wehave n )]TJ/F11 9.9626 Tf 10.046 0 Td [(k )]TJ/F8 9.9626 Tf 10.046 0 Td [(2 p )]TJ/F11 9.9626 Tf 10.046 0 Td [(` 0.Allthatremainsisdeterminingthesign oftheremainingpart, n n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(p +2 ` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k )]TJ/F10 6.9738 Tf 6.226 0 Td [(p +2 ` )]TJ/F1 9.9626 Tf 9.963 14.048 Td [( n p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q p : Case1 .If p n= 2,then n n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(p +2 ` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k )]TJ/F10 6.9738 Tf 6.227 0 Td [(p +2 ` = n p + k )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ` )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 q p + k )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 ` ; andsince2 ` k wehavebyassumptionthat )]TJ/F17 7.9701 Tf 17.786 -4.379 Td [(n p + k )]TJ/F15 7.9701 Tf 6.586 0 Td [(2 ` )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 q p + k )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 ` )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(n p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q p ,thusthecontributions ofthesetwotermsto.3isnonnegative. Case2 .If p>n= 2,then p n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k +2 ` = 2implies n= 2 p n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(p +2 ` ,and n n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(p +2 ` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k )]TJ/F10 6.9738 Tf 6.227 0 Td [(p +2 ` )]TJ/F1 9.9626 Tf 9.962 14.048 Td [( n p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 q p 0 followsbythesymmetryofthedistribution. Allothertermscontributezero.Therefore,wehaveshownthat.3willbesatised,andthe proofiscomplete. Remark .Considerthedistributionforwhichtheurnscontaining n> 1+1"ballsor n )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"balls haveprobabilityzero,withallotherurnsequallylikely.Then,iftherst n )]TJ/F8 9.9626 Tf 10.13 0 Td [(1drawsresultinall +1"balls, P n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =1 Y i = n )]TJ/F8 9.9626 Tf 10.288 0 Td [(1 0,buttheplayerwouldknowthelastballis )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"andnotplace abet.Similarly,weightingtheprobabilitiessothatcloser-to-neutralurnsaremorelikelyproduce apointofdiminishedreturns-if P k i =1 Y i becomestoolarge,thesumismorelikelytodecreasea sell-o,andtheplayerwillstopbetting,andifthesumbecomeslargeonthenegativeside,the sumismorelikelytoincreasearally,andtheplayerwillstarttobet. Wehavethetoolstogiveanexplicitformulafor G n; ;1 ; 1,where satisestheconditionsof Lemma7.4,basedonpropertiesofvarious n )]TJ/F11 9.9626 Tf 10.337 0 Td [(p;p ;1 ; 1urns.RecallfromTheorem3.2thatfor m p G m;p ;1 ; 1= )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(m + p i .For0 p n ,denote G n;p = E )]TJETq1 0 0 1 253.064 181.084 cm[]0 d 0 J 0.398 w 0 0 m 7.833 0 l SQBT/F11 9.9626 Tf 253.064 172.881 Td [(G n; ;1 ; 1 j thereare p +1"ballsinitially : Foranyinitialpriordistribution ,thetotalprobabilitytheoremgivesusthat .7 G n; ;1 ; 1= n X j =0 q j G n;j : 114

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If satisestheconditionsofLemma7.4,then .8 G n;p = G 2 p )]TJ/F10 6.9738 Tf 6.227 0 Td [(n n )]TJ/F11 9.9626 Tf 9.963 0 Td [(p;p ;1 ; 1 ; 0 p n: Theorem7.5. If isadistributionsatisfyingtherequirementsofLemma7.4,then G n; ;1 ; 1= n X j =0 q j G n;j ; where .9 G n;j = 8 > > > > < > > > > : )]TJ/F1 9.9626 Tf 7.749 14.047 Td [( n j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 j X i =0 n i ; if 0 j 2 j )]TJ/F10 6.9738 Tf 6.227 0 Td [(n n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j;j ;1 ; 1=2 j )]TJ/F11 9.9626 Tf 9.963 0 Td [(n )]TJ/F11 9.9626 Tf 9.962 0 Td [(G n )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 j j;n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j ;1 ; 1 = j )]TJ/F11 9.9626 Tf 9.963 0 Td [(n )]TJ/F11 9.9626 Tf 9.963 0 Td [(G j;n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j ;1 ; 1= j )]TJ/F11 9.9626 Tf 9.963 0 Td [(n )]TJ/F1 9.9626 Tf 9.963 14.048 Td [( n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 n i : Havingshownthat.9holds,weusethetotalprobabilityruletocompletetheproof. BoththeuniformandbinomialdistributionssatisfytherequirementsofLemma7.4.When is uniform,wehavethefollowingresult. Corollary7.6. Suppose istheuniformdistributionon f 0 ; 1 ;:::;n g 1 If n isodd,then n +1 G n; ;1 ; 1= n 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b n= 2 c X j =1 j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 n i n j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : 115

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2 If n iseven,then n +1 G n; ;1 ; 1= n 2 +2 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n n= 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 n= 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X j =1 j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 n i n j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : TheformulasofCorollary7.6abovegive G ; ;1 ; 1=0 ; G ; ;1 ; 1= 1 6 ; G ; ;1 ; 1= 1 3 ; G ; ;1 ; 1= 8 15 ; G ; ;1 ; 1= 11 15 ; G ; ;1 ; 1= 199 210 ; ofwhichwenotethatthecases n =1 ; 2 ; 3 ; giveduplicateresultstothoseof[7],withtheresult for n =4correctingaminorerror.Weshallgiveavertical"formulaholdingforgeneral s and t ,tocomparewiththehorizontal"formulaofCorollary7.6,withTheorem7.13.Sincewehave asymptoticknowledgeathandforthepresentationofCorollary7.6,wecanquicklyassessthe asymptoticvalueof G n; ;1 ; 1forlarge n Thesum P p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(n i )]TJ/F17 7.9701 Tf 10.959 -4.379 Td [(n p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 isthevaluefora n )]TJ/F11 9.9626 Tf 10.356 0 Td [(p;p ;1 ; 1urnwith p n= 2usingtheoriginal sumforthecase p = n= 2,weknowthatas n !1 thesumis O p p = O p n .Sincewehave n +1suchsums,theirtotalcontributionto n +1 G n; ;1 ; 1is O n 3 = 2 .Therefore,as n !1 .10 G n; ;1 ; 1 n 4 : Weshallndabetterboundforthebig O termwithTheorem7.13. Remark .InChen etal. [8,Theorems5and6],itwasshownthattherandomstoppingurn withaninitialdistributionthatisuniformalsohasanasymptoticvalueof n= 4whenanoptimal strategyisused.Therefore,thatstrategyisdesignedtotrytocapturetheinitialweightoftheurn ifitispositiveandstopbeforetoolongiftheinitialweightisnegative.Itshouldbenoted,however, thatiftheplayerusesa k intheholestrategy"with,say, k = An 3 = 4 ,thentheasymptoticvalueof theexpectedgainwiththisstrategywillstillbe n= 4,astheaverageofthepositiveinitialweightsis stillfarlargerthan k ,andforthemajorpositivecontributorsitisextremelyunlikelytheplayer willget k inthehole." As G n;n ;1 ; 1= p n= 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o = o n as n !1 G n; ;1 ; 1 n= 2islargerthan G n;n ;1 ; 1forlarge n ,thoughthisisnotthecasefor n =1 ; 2 ; 3. n =4isthesmallest n forwhich 116

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theBayesianvalueexceedsthevalueofitscorrespondingneutralurn,as G ; ;1 ; 1=436 = 315 > 93 = 70= G ; 4;1 ; 1. TheBayesianurnwith n ballsgenerallygainsmoreontheaveragethanthecorrespondingneutral urnon n balls,byalargemargin.Whythishappensisclear-thereisroughlya1 = 4chancethatthe initialweightoftheurnisatleast n= 2,puttingthegainsfromthoseurnstobelargerthat n= 8,with thelossesassociatedwiththoseurnsnotenoughtomakeadent.Abettercomparisonto G n; ;1 ; 1 with uniformistheaverageof all ofthevaluesoverurnswith n balls,i.e. 1 n +1 P n j =0 G j;n )]TJ/F11 9.9626 Tf 8.378 0 Td [(j ;1 ; 1. Thissumisclearlybiggerthan G n; ;1 ; 1,aseachurn'soptimalstrategywillgaintheinitialweight oftheurnifitispositive,plusabitmore.Thisaverageisalsoasymptotically n= 4,asthetheextra gainsfromcrossingsaresurely O p n Weseekadistribution ^ thatgivesavalue G n; ^ ;1 ; 1comparableto G n;n ;1 ; 1forlarge n SinceweonlyhaveanexplicitformulafordistributionssatisfyingtheconditionsofLemma7.4,we shallassumethatthoseconditionsneedtobemet.Luckilyenough,thereisadistributionsatisfying thoseconditionsthatgivesanexpectedgainofzero. Theorem7.7. Suppose isthebinomialdistributionwith q j = )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(n j 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(n for 0 j n .Thenthe n; ;1 ; 1 urnisafairgame,thatis, G n; ;1 ; 1=0 WeshallgiveaproofundermoregeneralcircumstanceswithTheorem7.11.Itshouldbeclear that satises.5.With q j = )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(n j 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(n ,eachrealizationon n ballsconsistingof )]TJ/F8 9.9626 Tf 7.748 0 Td [(1"sand+1"s isequallylikely.Then,regardlessofwhathasbeendrawnfromtheurn,the k th ballisequallylikely tobe )]TJ/F8 9.9626 Tf 7.748 0 Td [(1"or+1,"andtheexpectedgainifabetisplacediszero.Therefore,anystrategyis optimal,includingthestrategyindicatedby.4. Withtheuniformdistributiongivingavaluetoohigh,andthebinomialdistributiongivinga valuetoolow,ahybriddistributionofthetwoshouldgiveuswhatwewant.For 2 [0 ; 1],let .11 q j = 1 n +1 ;q 0 j = n j 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(n ; ^ q j = q j + )]TJ/F11 9.9626 Tf 9.963 0 Td [( q 0 j : Considertheinitialdistributions = f q j g n j =0 0 = f q 0 j g n j =0 ,and ^ = f ^ q j g n j =0 .Itisclearfrom.7 that G n; ^ ;1 ; 1= G n; ;1 ; 1+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( G n; 0 ;1 ; 1 n= 4 ; as n !1 ; 117

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using.10andTheorem7.7.Taking = An )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 ,with A = p 2 wehaveforlargeenough k that G k; ^ ;1 ; 1 G k;k ;1 ; 1. Areversaloftheinequalityof.5givesasecondfamilywithasimpliedbettingrule. Lemma7.8. Supposetheprobabilitydistribution = f q j g n i =0 satises q j = q n )]TJ/F10 6.9738 Tf 6.226 0 Td [(j .Thenanoptimal bettingstrategyforthe n; ;1 ; 1 urnis .12 betonthe k +1 th ballifandonlyif k X i =1 Y i 0 ifandonlyif 0 j 1 j 2 n= 2 implies .13 n j 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 q j 1 n j 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 q j 2 : ThedistributionscoveredbyLemma7.8mightbeconsideredmorepractical,astheweights aremoreconcentratedatthecenter,whereasthedistributionscoveredbyLemma7.4favoredthe extremeends.Withthecenter-favoreddistribution,capturingtheinitialweightbecomessecondary tothegainsmadefromtheuctuationsalongtheway.Thus,theoptimalstrategyismoreinline withoptimalstrategiesassociatedwiththenonrandom m;p ;1 ; 1urns.Underthecircumstances ofLemma7.8,giventheurncontains j +1"balls,usinganoptimalstrategytheplayerwillgain G p )]TJ/F10 6.9738 Tf 6.226 0 Td [(m m;p ;1 ; 1.Thereforewehave: Theorem7.9. If isadistributionsatisfyingtherequirementsofLemma7.8,then G n; ;1 ; 1= n X j =0 q j G n;j ; where .14 G n;j = 8 > > > > < > > > > : j )]TJ/F11 9.9626 Tf 9.963 0 Td [(n + n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 n i ; if 0 j n= 2 ; n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n )]TJ/F10 6.9738 Tf 6.226 0 Td [(j X i =0 n i ; if n= 2
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algorithmTheorem1.1givesusanalgorithmthatcalculates G n; ;1 ; 1.Theweaknessofthis algorithmisinthenumberofcalculationsrequiredtocalculatetheexpectedgain,asnotedin[8]. Theorem7.10. Givenan n; ;1 ; 1 urn,for 0 j n let P j = q j )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(n j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 and a n;j = c n;j = 0 .For 0 j n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ,let a n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;j = P j +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(P j ,andfor 0 i n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 andeach j let a i;j = a i +1 ;j + a i +1 ;j +1 : For 0 i n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 andeach j let c i;j =max f 0 ;a i;j g + c i +1 ;j + c i +1 ;j +1 : Thenanoptimalbettingpolicyformaximizingtheexpectedgainis:if k ballshavebeendrawn, ` of themplus,acceptthenextballifandonlyif a k;` > 0 .Thevalue G n; ;1 ; 1 oftheurnunder optimalplayis c ; 0 Figure7.1. a i;j and P j for n =4and uniform. Theprobabilisticstructureofthealgorithmisexactlythesameaswiththeoriginalstopping urn.See[4,Appendix2]forthedetails.Thistime,sincewecanstopandstartasmanytimesas necessary,thedecisiontoplaceabetisbasedentirelyonthenextdraw,whereaswiththestopping urn,theplayermightbewillingtotakeaexpectedlossonthenextballinthehopesofalater rebound.Theresultsforthisalgorithmwith n =4and uniformaregiveninFigures7.1and 7.2.Thealgorithmgives G ; ;1 ; 1=32 = 60=8 = 15,matchingtheoutputoftheformulagivenby Corollary7.6.Inthistriangulartableandtheonestofollow,thetopboxinthecolumncorresponds toallballsbeing+1,"anddecreasingonthewaydown. 119

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Figure7.2. c i;j for n =4and uniform.Thevalueofthe ; ;1 ; 1urnis c ; 0=8 = 15. InFigure7.3,weseethatwhentheconditionsofLemma7.8aremet,theoptimalbettingpolicy istheopposite-betif P k j =1 Y k 0,"apolicyindicatingexpectarally,anticipateasello." Figure7.3. P j and a i;j fortheinitialdistribution with p 0 = p 4 =1 = 20, p 1 = p 3 =1 = 4,and p 2 =2 = 5. 7.5. AFair n; ; s;t Urn. Wehavenotedthatif s = t =1and isthebinomialdistribution, then G n; ;1 ; 1=0.Thisisalsothecasewiththe n; ; s;t urns. Theorem7.11. Suppose isthebinomialdistributionwithparameter n andprobabilityofsuccess s= t + s .Thatis,theprobability q j theurncontains j + t "ballssatises q j = n j s t + s j t t + s n )]TJ/F10 6.9738 Tf 6.227 0 Td [(j : Thenthe n; ; s;t randomacceptanceurnisafairgame,thatis, G n; ; s;t =0 : 120

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Proof. Supposethat k ballshavebeendrawn,with ` ofthem+ t ."Weshowthat.3always returnszeroif0 ` k ,beginningbydiscardingtermsthatdonotcontributetothesum. n X j =0 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` s t + s j t t + s n )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` t + s )]TJ/F8 9.9626 Tf 9.963 0 Td [( n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = t k )]TJ/F10 6.9738 Tf 6.226 0 Td [(` s ` t + s k n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k + ` X j = ` n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k j )]TJ/F11 9.9626 Tf 9.963 0 Td [(` s t + s j )]TJ/F10 6.9738 Tf 6.227 0 Td [(` t t + s n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k )]TJ/F10 6.9738 Tf 6.226 0 Td [(j + ` )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( j )]TJ/F11 9.9626 Tf 9.962 0 Td [(` t + s )]TJ/F8 9.9626 Tf 9.963 0 Td [( n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k : Replacing j )]TJ/F11 9.9626 Tf 9.997 0 Td [(` with j ,and n )]TJ/F11 9.9626 Tf 9.997 0 Td [(k with n ,weseethefamiliarformsofabinomialdistribution,and thesummationneglectingtheoutsidefactorbecomes,usingthisnewbinomialdistribution, n X j =0 n j s t + s j t t + s n )]TJ/F10 6.9738 Tf 6.227 0 Td [(j )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(j t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(sn = t + s E [ j ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(sn =0 : Therefore,thesumvanishesfornonnegativeintegers ` k .Thisimpliesthateitheroption,passor bet,isoptimalateverystage.Therefore, G n; ; s;t =0. 7.6. The n; ; s;t Urnswith Uniform. Nowsuppose isuniform,thatis, q p = urnhas p + t "balls= 1 n +1 ; 0 p n: Wearetoplaceabetifandonlyif Y k +1 = t j y 1 ;y 2 ;:::;y k s= s + t .Suppose y 1 ;:::;y k are given,andthat ` ofthemare+ t "balls.Givenaparticularsequenceof k balls, ` ofwhichare+ t ," wehave Y k +1 = t j y 1 ;:::;y k = Y 1 = y 1 ;:::;Y k = y k ;Y k +1 = t Y 1 = y 1 ;:::;Y k = y k ; andwecancalculatethetwoprobabilitiesontheright-handsideexplicitly.For0 j n Y 1 = y 1 ;:::;Y k = y k j j = j ` n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j k )]TJ/F10 6.9738 Tf 6.226 0 Td [(` n k ; thus Y 1 = y 1 ;:::;Y k = y k = n X j =0 j ` n )]TJ/F11 9.9626 Tf 9.962 0 Td [(j k )]TJ/F10 6.9738 Tf 6.227 0 Td [(` n +1 k +1 = ` k )]TJ/F11 9.9626 Tf 9.963 0 Td [(` k +1! n +1 k +1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 n X j =0 j ` n )]TJ/F11 9.9626 Tf 9.962 0 Td [(j k )]TJ/F11 9.9626 Tf 9.962 0 Td [(` = ` k )]TJ/F11 9.9626 Tf 9.962 0 Td [(` k +1! ; thelastfollowingfromtheconvolutionidentity[10,eq..26].Therefore,wehave .15 Y k +1 = t j y 1 ;:::;y k = ` +1! k )]TJ/F11 9.9626 Tf 9.962 0 Td [(` = k +2! ` k )]TJ/F11 9.9626 Tf 9.962 0 Td [(` = k +1! = ` +1 k +2 : 121

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Thereforeanoptimalpolicyis:Iftheplayerhasseen k balls, ` ofwhichare+ t ,"thentheplayer shouldacceptthenextballifandonlyif ` +1 k +2 s t + s t ` +1 s k +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(` .16 t` )]TJ/F11 9.9626 Tf 9.962 0 Td [(s k )]TJ/F11 9.9626 Tf 9.963 0 Td [(` s )]TJ/F11 9.9626 Tf 9.963 0 Td [(t: Thusanoptimalstrategyistohavetheplayerbetifandonlyif P k i =1 Y i s )]TJ/F11 9.9626 Tf 9.78 0 Td [(t .Thiscondition isconsistentwiththatgivenin[7]forthecase s = t =1.Thus,if t =1and s =100,aplayerwould behesitanttobeteveniftherst98ballswere+1",whileontheotherhand,if t =100and s =1, thenourplayerwouldnotbediscouragedeveniftherst98ballswere )]TJ/F8 9.9626 Tf 7.749 0 Td [(1."Notethatif t ns thenthelowestvalue t` )]TJ/F11 9.9626 Tf 10.218 0 Td [(s k )]TJ/F11 9.9626 Tf 10.218 0 Td [(` cantakewith k > < > > : t )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 2 ; if t s 0 ; if t
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As n becomeslarger,ageneralformulafor G n; ; s;t becomesincreasinglydependenton s and t asthebettinglinecanappearinmoreplaces: G ; ; s;t = 8 > > > > > > > > > > < > > > > > > > > > > : t )]TJ/F11 9.9626 Tf 9.962 0 Td [(s; if t 2 s 5 t )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 s 6 ; if s t< 2 s 2 t )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 6 ; if s 2 t > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : 3 t )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 2 ; if t 3 s 17 t )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 s 12 ; if2 s t< 3 s 15 t )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 s 12 ; if s t< 2 s 7 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 s 12 ; if s 2 t
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Proof. Withaninitialdistributionthatisuniform,givenaspecicorderingof k balls, ` ofthem + t ,"wesawthat Y 1 = y 1 ;:::;Y k = y k = ` k )]TJ/F11 9.9626 Tf 9.963 0 Td [(` k +1! : Sincethereare )]TJ/F17 7.9701 Tf 5.48 -4.379 Td [(k ` suchsequences,wehave f Y 1 ;:::;Y k g = f + t;:::; + t | {z } ` times ; )]TJ/F11 9.9626 Tf 7.749 0 Td [(s;:::; )]TJ/F11 9.9626 Tf 7.749 0 Td [(s | {z } k )]TJ/F10 6.9738 Tf 8.041 0 Td [(` times g = 1 k +1 : Given ` oftherst k ballsare+ t ,"theexpectedgainonthenextballis max 0 ; 1 k +2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(t ` +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(s k +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(` : Thus,wesumoverallpossible ` and k .When s = t =1,sincewebetifandonlyif ` k= 2,we havethat G n; ;1 ; 1= n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 X k= 2 ` k 2 ` )]TJ/F11 9.9626 Tf 9.962 0 Td [(k k +1 k +2 : Toderivetheformulainthetheorem,weobserverstthatfor k 0, X k= 2 ` k 2 ` )]TJ/F11 9.9626 Tf 9.962 0 Td [(k k +1 k +2 = 8 > > < > > : 1 4 )]TJ/F8 9.9626 Tf 26.361 6.739 Td [(1 4 k +1 ; if k iseven ; 1 4 )]TJ/F8 9.9626 Tf 26.361 6.739 Td [(1 4 k +2 ; if k isodd : Itnowfollowsthat n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 X k= 2 ` k 2 ` )]TJ/F11 9.9626 Tf 9.963 0 Td [(k k +1 k +2 = n +1 4 )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 b n= 2 c X ` =0 1 2 ` +1 + n 4 n +1 = n +1 4 )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 H n + 1 4 H )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [(b n= 2 c )]TJ/F11 9.9626 Tf 23.263 6.74 Td [( n 4 n +1 : Ifwewantaformfor G n; ; s;t similartothatofCorollary7.6,wenotethat E G n; ; s;t j thereare p + t "balls = G X 0 + t )]TJ/F10 6.9738 Tf 6.227 0 Td [(s n )]TJ/F11 9.9626 Tf 9.963 0 Td [(p;p ; s;t ; where X 0 p = p t + s )]TJ/F11 9.9626 Tf 9.962 0 Td [(ns .Thus G n; ; s;t = 1 n +1 n X j =0 G X 0 j + t )]TJ/F10 6.9738 Tf 6.227 0 Td [(s n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j;j ; s;t : 124

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Theoptimalstrategyaimstocapture X 0 j + t )]TJ/F11 9.9626 Tf 10.08 0 Td [(s ifitisnonnegative.Someofitmaybemissed duetocrossings.Likethe s = t =1urn,theplayerwillsuerlossesalongthewayinaneortto securethispossiblylargepositivegain. With s = t =1,wesawthattheasymptoticvalueof G n; ;1 ; 1was n= 4withtheuniform distribution.Wecanalsocalculatetheasymptoticvalueoftheexpectedgainwith uniform,for the n; ; s;t urns. Theorem7.13. Supposethat isuniformon f 0 ;:::;n g .Thenas n !1 ,wehave G n; ; s;t = n 2 t 2 t + s + O ln n : Proof. For0 ` k ,let f `;k =max 0 ;t ` +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(s k +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(` ; andlet L 0 = ks t + s + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(t t + s ;L = d L 0 e ; = L )]TJ/F11 9.9626 Tf 9.963 0 Td [(L 0 : For `>L 0 wehave f `;k > 0.Observethatfor ` L 0 f ` +1 ;k = f `;k + t + s .Then n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k X ` =0 f `;k k +1 k +2 = t + s n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 1 k +1 k +2 k )]TJ/F11 9.9626 Tf 9.963 0 Td [(L +1 + k )]TJ/F11 9.9626 Tf 9.962 0 Td [(L k )]TJ/F11 9.9626 Tf 9.963 0 Td [(L +1 2 = O ln n + t + s 2 n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 k )]TJ/F11 9.9626 Tf 9.962 0 Td [(L k )]TJ/F11 9.9626 Tf 9.963 0 Td [(L +1 k +1 k +2 = O ln n + t + s 2 n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 k )]TJ/F11 9.9626 Tf 9.963 0 Td [(L 0 2 k +1 k +2 : Since k )]TJ/F11 9.9626 Tf 9.962 0 Td [(L 0 = kt= t + s + O ,wethenhave = O ln n + t 2 2 t + s n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 k 2 k +1 k +2 = O ln n + n 2 t 2 t + s ; completingtheproof. Remark. IfourplayerusesthecorrespondingstrategytothatofLemma7.4,bettingifandonlyif P k i =1 Y i 0,thenourplayerwouldbetifandonlyif ` ks= t + s .Therefore,since s;t 0 125

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thedierencebetweenthisstrategyandtheoptimaloneislessthanaball'sdierence.Infact,the optimalstrategywillcaptureabitlessofthosepositiveinitialvaluesupto2 t + s duetocrossings if s>t ,whileitmightpickasmallnegativeinitialvalueif t>s .Theplayerusingtheoptimal strategywillnotloseasmuchalongtheway,whichmorethanmakesupthedierence. 7.7. AnAlgorithmfortheGeneralRandomUrn. Asforanalgorithmforcalculatingthe value G n; ; s;t inthemannerofTheorems1.1and7.10,theunderlyingprobabilisticstructure doesnotchangewhentheweightsoftheballschange.Thedierenceisinhowthe b i;j 'sTheorem 1.1and c i;j 'sTheorem7.10arecalculated.If q i;j istheprobabilityofaspecicsequenceof i balls,with j ofthem+ t ,"then )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(see[4,Appendix2] q i;j = q i +1 ;j +1+ q i +1 ;j ; andgiventhatsequence,theprobabilitythenextballisa+ t "is q i +1 ;j +1 =q i;j andthe probabilityitisa )]TJ/F11 9.9626 Tf 7.748 0 Td [(s "is q i +1 ;j =q i;j .Thustheexpectedgainforthenextdrawwith s = t =1 equals q i +1 ;j +1 q i;j )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(q i +1 ;j q i;j = a i;j q i;j : The q i;j inthedenominatorcanbedispatchedwith,thuswedonotneedtogothroughthe Figure7.5. a i;j and P j for n =3and uniform. unappealingtaskofcalculatingallofthetransitionprobabilities.Unfortunately,forthegeneral stoppingoracceptance n; ; s;t urn,theexpectedgainonthenextdrawwillbe t q i +1 ;j +1 q i;j )]TJ/F11 9.9626 Tf 9.962 0 Td [(s q i +1 ;j q i;j ; 126

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thusitappearsthatsometransitionprobabilitiesneedtobecalculatedinorderforsuchanalgorithm towork.Theonlyquestioniswhetheronlysomecanbecalculated.Wehave a i;j = q i +1 ;j +1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(q i +1 ;j ; and Figure7.6. c i;j ,^ c i;j smallertableatright,and q i; 0, i =1 ; 2 ; 3for s =2, t =3, n =3,and uniform.Thevalueofthe ; ;2 ; 3urnis c ; 0=23 = 12. t q i +1 ;j +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(s q i +1 ;j = s )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(q i +1 ;j +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(q i +1 ;j + t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s q i +1 ;j +1 ; .17 = s a i;j + t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s q i +1 ;j +1 ; andwecanreduce q i +1 ;j +1to q i +1 ;j +1= q i +1 ; 0+ j X k =0 a i;k : Therefore,onlythecollection f q i; 0 g n i =1 theprobabilitiesthattherst i ballsareall )]TJ/F11 9.9626 Tf 7.748 0 Td [(s ,"1 i n wouldneedtobeknown. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(Knowing f q i;i g n i =1 alsoisgoodenough. Thisisanimprovement comparedtothetotalnumberoftransitionprobabilities,whichisroughly n 2 = 2.Then,deningfor 0 i n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(again, c n;j =0forall j ^ c i;j = s a i;j + t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s q i +1 ; 0+ j X k =0 a i;k # ; c i;j =max 0 ; ^ c i;j + c i +1 ;j + c i +1 ;j +1 ; oneshouldbetif^ c i;j > 0andpasswhen^ c i;j < 0.Thevalue G n; ; s;t wouldequal c ; 0. With uniform,wehavefrom.15that q i; 0=1 = i +1for0 i n )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(Since q i;j =1 = i +1for 0 j i wedonotactuallyneedtogothroughthereductiontotheoutsidetransitionprobabilities; 127

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Figure7.7. Thealgorithmappliedtothecase s =9, t =4, n =3and uniform. Wehave G ; ;9 ; 4=3 = 12=1 = 4. wecanworkdirectlyfrom.17. Figure7.5givesthestructureforthecase n =3and uniform. Theresultofthealgorithmwith s =2, t =3, n =3,and uniform,isgiveninFigure7.6,andthe resultwith s =9, t =4, n =3and uniformisgiveninFigure7.7.Thelatterexampleshowsthat ^ c i;j 6 =^ c i +1 ;j +1+^ c i +1 ;j asitappearedtobeintheformerexample. Theweaknessofsuchanalgorithm,again,isthenumberofcalculationsneededtocalculatethe naloutput.Sincesometransitionprobabilitiesareneeded )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(nottomentionusingvarioussumsof the a i;j 'stoshortenthelistoftransitionprobabilitiesneeded ,thesealgorithmsareevenmore cumbersome.WepresentthealgorithmformallywithTheorem7.14. Theorem7.14. Givena n; ; s;t urn,for 0 j n let P j = q j )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 and a n;j = c n;j =0 For 0 j n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ,let a n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;j = P j +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(P j ,andfor 0 i n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 andeach j let a i;j = a i +1 ;j + a i +1 ;j +1 : For 0 i n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 andeach j let ^ c i;j = s a i;j + t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s q i +1 ; 0+ j X k =0 a i;k # ; c i;j =max 0 ; ^ c i;j + c i +1 ;j + c i +1 ;j +1 ; where q i +1 ; 0 isexplicitlygivenby q i +1 ; 0= n X j = i +1 j i +1 n i +1 q n )]TJ/F10 6.9738 Tf 6.227 0 Td [(j : 128

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Thenanoptimalbettingpolicyformaximizingtheexpectedgainis:if k ballshavebeendrawn, ` of themplus,acceptthenextballifandonlyif c k;` > 0 .Thevalueoftheurnunderoptimalplayis c ; 0 Wealsoremarkthat,withafewadjustments,analgorithmgivingthevalueofarandomgeneralizedstoppingurncanbegivenaswell.Weomitthedetails. 129

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8. ARuinProblem Thefamilyofoptimalbettingstrategiesforthe m;p ; s;t urnnotonlyguaranteesanonnegative value,butalsoanonnegativegainregardlessofhowtheballsendupbeingdrawnfromtheurn. However,whilethegameisinprogress,ourplayermayndhimselfdigginginhisownpocketsin ordertocontinue.Thequestionathandhereisthis:Howmuchcapitalmightoneneedtoplay thegameusinganyoftheoptimalstrategies?Wecananswerthisquestioninthecase s = t =1, thankstothereectionmethod.Answeringthisquestionundermoregeneralcircumstancesisan openproblem.Instead,weattempttogiveageneralideaofwhatisneededwhen m and p arelarge. Werststatetheproblemforthenonrandomurn. 8.1. TheRuinProblemforthe m;p ; s;t Urn. Wegivetheplayera bank b = B 0 ,andmonitor itssizeasthegameprogresses.Thatis,for1 n m + p : B n = 8 > > > > > > < > > > > > > : B n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + t; ifabetisplacedonthe n th ballanditisa+ t ," B n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(s; ifabetisplacedonthe n th ballanditisa )]TJ/F11 9.9626 Tf 7.749 0 Td [(s ," B n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ; ifnobetisplaced. Wesaythataplayeris ruined if B n max f 0 ;pt )]TJ/F11 9.9626 Tf 9.542 0 Td [(ms g + b )]TJ/F11 9.9626 Tf 9.542 0 Td [(s .Weusethisconditiontoshowthatif b islargecomparedto p p ,the probabilityofruingoestozeroas p goestoinnity.However,giventhatundermostcircumstances, G m;p ; s;t = o b ,fromapracticalstandpointthegameprobablywouldnotbeplayedconsidering thecapitalbroughttothetable. 130

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8.1.1. TrendsandPreliminariesforthe m;p ;1 ; 1 and m;p ;1 ;t Urns. Wehavemanymoretools atourdisposalwhen s = t =1.Infact,wehaveenoughtogiveasolutiontotheruinproblem, thoughitisquitecomplicated.Wesolvetheproblemforthe m;p ;1 ; 1urninthenextsection. Here,wesetuppreliminariesandgivegeneraltrends.Someofthecomingresultsholdforthe m;p ;1 ;t urnsaswell,sowepresentthoseresultsunderthemoregeneralcircumstances. ByLemma3.5,withbank b wehavethat P X n = b forsome n = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p )]TJ/F11 9.9626 Tf 9.963 0 Td [(b ; if m p and P X n = p )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + b forsome n = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p + b ; if p m Usingtheseresults,wecannowshowsometrends: Lemma8.1. Supposetheplayerusesanoptimalbettingstrategy.If p=m > 1 ,then,as p !1 theprobabilityofruinwithbank b isboundedaboveby )]TJ/F10 6.9738 Tf 6.227 0 Td [(b .If m=p > 1 ,then,as m !1 ,the probabilityofruinwithbank b isboundedaboveby )]TJ/F10 6.9738 Tf 6.227 0 Td [(b Proof. For p>m and p=m > 1,theprobabilityofruinisboundedaboveby m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p + b = m b p + b b m p b )]TJ/F10 6.9738 Tf 6.227 0 Td [(b : Theothercaseissimilar,andtheproofisomitted. Lemma8.2. If p m ,and h = p )]TJ/F11 9.9626 Tf 10.055 0 Td [(m ,thentheprobabilityofruinwithbank b isatmost e )]TJ/F10 6.9738 Tf 6.227 0 Td [(hb=p If m p ,and h = m )]TJ/F11 9.9626 Tf 9.962 0 Td [(p ,thentheprobabilityofruinwithbank b isatmost e )]TJ/F10 6.9738 Tf 6.227 0 Td [(hb=m Proof. Suppose p m .Wehave m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m + p p + b m p b = 1 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(h p b exp )]TJ/F11 9.9626 Tf 8.944 6.74 Td [(hb p ; asdesired.Theothercaseissimilar,andtheproofisomitted. Lemma8.2showsthatifmax f m;p g = o )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(b j p )]TJ/F11 9.9626 Tf 10.436 0 Td [(m j ,thentheruinprobabilitytendstozeroas m + p becomeslarge. 131

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Lemma8.3. Supposetheplayerusesanoptimalstrategy,andassumethat b = p p ,with > 0 If j p )]TJ/F11 9.9626 Tf 10.411 0 Td [(m j = p p ,thentheprobabilityofruinisboundedaboveby )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ o exp )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F11 9.9626 Tf 7.749 0 Td [( + ,as p !1 Proof. If p m ,thentheprobabilityofruinisboundedaboveby m + p p + b m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = m p b Y b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 1 )]TJ/F11 9.9626 Tf 13.816 6.74 Td [(i m Y b i =1 1+ i p : Since m = p )]TJ/F11 9.9626 Tf 9.962 0 Td [( p p and b = p p ,wehave m p b = 1 )]TJ/F11 9.9626 Tf 15.163 6.739 Td [( p p p p e )]TJ/F10 6.9738 Tf 6.227 0 Td [( : Also, b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Y i =0 1 )]TJ/F11 9.9626 Tf 13.816 6.739 Td [(i m exp )]TJ/F8 9.9626 Tf 10.827 6.739 Td [(1 m b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 i =exp )]TJ/F11 9.9626 Tf 10.604 6.739 Td [(b b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 m ; and b Y i =1 1+ i p exp 1 p b X i =1 i )]TJ/F8 9.9626 Tf 15.899 6.74 Td [(1 2 p 2 b X i =1 i 2 =exp b b +1 2 p )]TJ/F11 9.9626 Tf 11.159 6.74 Td [(b b +1 b +1 12 p 2 : Substitutinginfor b and m ,anupperboundforthequotientofthetwoproductsis exp )]TJ/F11 9.9626 Tf 7.749 0 Td [( 2 + O p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 : Theupperboundfortheruinprobabilityisthusexp )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F11 9.9626 Tf 7.749 0 Td [( + + O p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 ,asdesired.Thecase with m p issimilarlyshown,onlythistimetherolesof m and p arereversedsince m p ,we have p m p p .Weomittheproof. Lemma8.3showsthatifweleteither or tendtoinnity,theprobabilityofruintendstozero. Uptothispoint,wehavebeentalkingonlyaboutruinwhileusingthezero-betstrategy.The necessaryconditionforruinwiththezero-betstrategyisalsonecessaryusinganyoptimalstrategy. Therefore,theresultsaboveapplywithanyoptimalstrategy.Asmentionedatthestartofthis chapter,theruinprobabilitiesdovaryaccordingtothechoiceofoptimalstrategy.Infact,wecan showthatthezero-passstrategyissaferthanthezero-betstrategyforthe m;p ;1 ; 1urns.This canbeshownforthe m;p ;1 ;t urns,sowepresentandprovethenextresultinthatcontext. 132

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Lemma8.4. Let r b m;p ;1 ;t and r 0 b m;p ;1 ;t denotetheprobabilityofruinonthe m;p ;1 ;t urn usingthezero-passstrategyandzero-betstrategy,respectively,withinitialbank b .Then r b m;p ;1 ;t r 0 b m;p ;1 ;t : Proof. Let S S 0 denotethecollectionofrealizationsfromthe m;p ;1 ;t urnforwhichruinoccursusingthezero-passzero-betstrategywithinitialbank b .Since r 0 b m;p ;1 ;t = j S 0 j )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 r b m;p ;1 ;t = j S j )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,itsucestoshow j S jj S 0 j fortherstinequality.Toshowthis,we shallprovideaninjectivemapfrom S into S 0 .Observethatsince X n > 0isequivalentto X n 1, wecanre-statethezero-passstrategyasbetifandonlyiftheurnweightisatleastone." First,suppose m pt .Let 2 S .Thenthereisaminimal N forwhich B N =0.Also,since X n b> 0holdsforsome n ,thereisa+1"tripfromtheinitialstartingposition.Wemap to therealization 0 asfollows:Shiftthe+1"tripfromthestartingposition,andmoveitsothatit followstherstpointofruin.Thatis,if = P 1 P 2 P 3 ,with P 1 a+1"tripand j P 1 P 2 j = N ,then 0 = P 2 P 1 P 3 .Clearly, 0 isarealizationfromthe m;p ;1 ;t urn.Figure8.1showsthemapping with m = p =7, t =1,and b =1. Wenextshow 0 isin S 0 .If j P 1 j = k ,then .1 X n 0 = X n + k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; 0 n N )]TJ/F11 9.9626 Tf 9.963 0 Td [(k: Figure8.1. Mappingofazero-passruinrealizationblacktoazero-betruin realizationshadowwith b =1onthe ; 7;1 ; 1urn.Thedashedlineindicatesthe +1"triptobemoved,whilethesolidlineindicatestheruinsequence. Therefore,inthisrange X n 0 0ifandonlyif X n + k 1.Thusthesequenceofbets/passes over P 2 doesnotchangeuponmapping to 0 .Therefore,itfollowsthat B N )]TJ/F10 6.9738 Tf 6.226 0 Td [(k =0,sinceno bettingoccurson P 1 in .Thus 0 2 S 0 .Furthermore,itfollowsfrom.1thatruinrstoccurs 133

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for 0 after N )]TJ/F11 9.9626 Tf 9.941 0 Td [(k ballshavebeendrawn.Weusethisfacttoshowthemapisinjective.If 0 2 S 0 ruinrstoccursafter N ballshavebeendrawn,andthereisa+1"tripfrom X N ,thenshifting that+1"triptothebeginningoftherealizationproducesauniquerealization in S .Thus,we concludethat j S jj S 0 j If mm P X n pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + b forsome n = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q m;p pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + b = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b m )]TJ/F10 6.9738 Tf 6.226 0 Td [(b =t c X k =0 b kt + k + b kt + k + b k m + p )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt )]TJ/F11 9.9626 Tf 8.855 0 Td [(k )]TJ/F11 9.9626 Tf 8.856 0 Td [(b p )]TJ/F11 9.9626 Tf 8.855 0 Td [(k ; whilefor m pt wehave P X n b forsome n = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q m;p b = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p X k = d b=t e m )]TJ/F11 9.9626 Tf 8.856 0 Td [(pt + b m + p )]TJ/F11 9.9626 Tf 8.855 0 Td [(k )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt + b m + p )]TJ/F11 9.9626 Tf 8.855 0 Td [(k )]TJ/F11 9.9626 Tf 8.856 0 Td [(kt + b p )]TJ/F11 9.9626 Tf 8.856 0 Td [(k kt + k )]TJ/F11 9.9626 Tf 8.856 0 Td [(b k : Forthe m;p ; s;t urns,calculatingthisprobabilityseemstobedicultingeneral.Itispossible tocalculate P max n X n = b )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(andthus, P X n b forsome n = P j b P max n X n = j viathe s;t -zero-gainandballotnumbers,butduetotheircomplexity,weshallomitthoseresults.As aresult,weareforcedtouseboundsthatarenottightingeneral.Nevertheless,when b islarge enoughwhencomparedwith p p ,wecanshowtheprobabilityofruintendstozero. Theorem8.5. If,as p !1 b p 1 = 2+ !1 ; forsome > 0 thentheprobabilityofruinforthe m;p ; s;t urnwithbank b tendstozero. 134

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Proof. Theprobabilityofruinisboundedaboveby P )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(X n > max f 0 ;pt )]TJ/F11 9.9626 Tf 9.963 0 Td [(ms g + b )]TJ/F11 9.9626 Tf 9.962 0 Td [(s : ByLemmas2.11and2.12wemayassume )]TJ/F8 9.9626 Tf 7.749 0 Td [( t + s b )]TJ/F11 9.9626 Tf 9.963 0 Td [(s forsome n s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =0 X n P X n = b )]TJ/F11 9.9626 Tf 9.963 0 Td [(i : Onceagain,foreaseofcalculation,weshallassumethat b isdivisibleby s ,andshowonlythe i =0 termissmall.Theothertermswillfollowsimilarly. Write b = a 0 s = ast t + s and p 0 = p=s .Then X n P X n = b = m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k k t + s + a 0 ks m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 0 m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 0 : Now k t + s + a 0 ks exp 1 12 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k t + s + a 0 + 1 12 ks +1 + 1 12 kt + a 0 +1 .2 t + s k t + s + a 0 +1 = 2 s ks +1 = 2 t kt + a 0 +1 = 2 s k + at 2 k )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(k + a t + s 1+ at k ks 1 )]TJ/F11 9.9626 Tf 31.757 6.74 Td [(as k + a t + s kt + a 0 ; and m + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 0 m )]TJ/F11 9.9626 Tf 9.962 0 Td [(kt )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 0 t + s m + p )]TJ/F10 6.9738 Tf 6.226 0 Td [(k t + s )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 0 +1 = 2 s p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks +1 = 2 t m )]TJ/F10 6.9738 Tf 6.227 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 0 +1 = 2 s p 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(at 2 p 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(p 0 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(a t + s .3 1 )]TJ/F11 9.9626 Tf 19.468 6.74 Td [(at p 0 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k p )]TJ/F10 6.9738 Tf 6.227 0 Td [(ks 1+ as p 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k )]TJ/F11 9.9626 Tf 9.962 0 Td [(a t + s m )]TJ/F10 6.9738 Tf 6.227 0 Td [(kt )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 0 exp 1 12 m + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t + s )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 0 + 1 12 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(ks +1 + 1 12 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(kt )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 0 +1 ; whilewestillhave .4 m + p p t + s m + p +1 = 2 s p +1 = 2 t m +1 = 2 r 1 2 p 0 exp 1 12 m + p +1 + 1 12 m + 1 12 p : 135

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Combining.2,.3,and.4reciprocals,theexponenton t + s is1 = 2,theexponenton s is )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2,andtheexponenton t is )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2,whilethetermswithintheexponentialfunctionarenegligible. Wenowfocusonthetwopairofoneplus"andoneminus"terms.Wehave 1+ at k ks 1 )]TJ/F11 9.9626 Tf 31.758 6.74 Td [(as k + a t + s kt + a 0 1 ; .5 1+ as p 0 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k )]TJ/F11 9.9626 Tf 9.963 0 Td [(a t + s m )]TJ/F10 6.9738 Tf 6.226 0 Td [(kt )]TJ/F10 6.9738 Tf 6.226 0 Td [(a 0 exp ast ; .6 1 )]TJ/F11 9.9626 Tf 19.468 6.74 Td [(at p 0 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k p )]TJ/F10 6.9738 Tf 6.226 0 Td [(ks exp )]TJ/F11 9.9626 Tf 7.749 0 Td [(ast )]TJ/F8 9.9626 Tf 17.389 6.74 Td [( at 2 s 2 p 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ; .7 withtheproductoftheright-handsidesof.5,.6,and.7equaling exp )]TJ/F11 9.9626 Tf 40.948 6.74 Td [(b 2 2 t + s 2 p )]TJ/F11 9.9626 Tf 9.962 0 Td [(ks : If b p 1 = 2+ forsome > 0noting p )]TJ/F11 9.9626 Tf 10.042 0 Td [(ks b=t ,thistermwillgotozerofasterthanthesquare roottermsof.2,.3,and.4willincrease.Thus,thesummandtendstozero.Similarly,the sumover k willcontribute O p terms,andtheexponentialissmallenoughthatthetotalsumwill alsotendtozero. Theruinprobabilitiesofsomeurnswith s =2, t =3,and b 6withthezero-betstrategyand thezero-passstrategyarerespectivelygiveninTables8.1and8.2. Table8.1. Some ; ;2 ; 3urnruinprobabilitiesinpercentwithvetotalballs andthezero-betstrategy. urn b =2 b =3 b =4 b =5 b =6 ; 0;2 ; 300000 ; 1;2 ; 32020000 ; 2;2 ; 36060303010 ; 3;2 ; 3504010100 ; 4;2 ; 32020000 ; 5;2 ; 300000 8.2. SolutionoftheRuinProblemforthe m;p ;1 ; 1 Urns. Themaindicultyinsolving theruinproblemisthattheceiling B n s israndom.Aspermanentgainsaremade,theplayer's pocketswilldeepen.However,wehaveseenwithTheorem4.18thatthereisarelationshipbetween thedistributionofthegainandthedistributionofthemaximumweightachievedbytheurn.Using 136

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Table8.2. Some ; ;2 ; 3urnruinprobabilitiesinpercentwithvetotalballs andthezero-passstrategy. urn b =2 b =3 b =4 b =5 ; 0;2 ; 30000 ; 1;2 ; 3202000 ; 2;2 ; 340401010 ; 3;2 ; 350401010 ; 4;2 ; 3202000 ; 5;2 ; 30000 thesamebasicprocedure,butthistimefocusedonlyonthegainsmade before thepointofruin,we cancalculatetheruinprobability. 8.2.1. RuinwiththeZero-BetStrategy. Letusbeginwiththezero-betstrategy.Given m p b ,and k ,dene E m;p b;k asthenumberofrealizationsfromthe m;p ;1 ; 1urnforwhichruinwilloccur withbank b ,andforwhichapermanentgainof k notincludingtheinitialweight p )]TJ/F11 9.9626 Tf 10.053 0 Td [(m ismade beforeruinoccurs.Thatis, E m;p b;k isthenumberofrealizationsforwhich B attainsamaximum valueof b + k beforehittingzero.Then,theprobabilityofruinwillbe r 0 b m;p ;1 ; 1= m + p p )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k E m;p b;k : Wecalculate E m;p b;k asaconvolutionoftwoobjects.Oneoftheseobjectsmaybecountedby thefollowingresult,foundinFeller[9,p.96,#3].Foraninformalproofviarepeatedreections,see S.G.Mohanty's LatticePathCountingandApplications [14,pp.6-7]. Lemma8.6. Feller Let a and b bepositive,and )]TJ/F11 9.9626 Tf 7.749 0 Td [(b
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Theorem8.7. Suppose k 0 .Let k = k + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + .Then E m;p b;k = p )]TJ/F10 6.9738 Tf 6.227 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(k X j = k m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 j )]TJ/F11 9.9626 Tf 9.962 0 Td [(b m )]TJ/F11 9.9626 Tf 9.962 0 Td [(j + k )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(S b;k +1 j + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( k S b;k j + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; where k =0 if k =0 andequals 1 otherwise. Proof. Suppose isarealizationforwhichtheplayerisruined,butmakesapermanentgainof k beforeruinoccurs.Weseparate intothreeparts: 1 shallbecomposedoftheballsbetuponupto andincludingthepointofruin,asintheproofofTheorem4.18; 2 issimilarto 1 ,butcontainsthe ballspassedon;and 3 istherealizationfollowingthepointofruin.Then 2 isa+ k + m )]TJ/F11 9.9626 Tf 9.599 0 Td [(p + trip, 3 isa )]TJ/F1 9.9626 Tf 7.749 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b + k + p )]TJ/F11 9.9626 Tf 9.994 0 Td [(m + "path,while 1 isapaththattakestheweight )]TJ/F11 9.9626 Tf 7.748 0 Td [(k )]TJ/F8 9.9626 Tf 9.994 0 Td [( p )]TJ/F11 9.9626 Tf 9.994 0 Td [(m + avoidstheweights )]TJ/F11 9.9626 Tf 7.749 0 Td [(k )]TJ/F8 9.9626 Tf 10.511 0 Td [(1 )]TJ/F8 9.9626 Tf 10.512 0 Td [( p )]TJ/F11 9.9626 Tf 10.512 0 Td [(m + and b ,andendsatweight b )]TJ/F8 9.9626 Tf 10.512 0 Td [(1,followedbya )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ball ruiningtheplayer.Thenumberofsuch 1 ,giventhatthereare j +1"balls j k in 1 ,equals S b;k +1 j + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( k S b;k j + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : When k =0,theconditionsofLemma8.6arenotmet.Concatenationof 2 and 3 givesapath from b to0thatreaches b + k + m )]TJ/F11 9.9626 Tf 10.331 0 Td [(p + ,containing p )]TJ/F11 9.9626 Tf 10.331 0 Td [(j +1"balls.ViaLemma3.4,wend thatthereare )]TJ/F17 7.9701 Tf 5.479 -4.379 Td [(m + p )]TJ/F15 7.9701 Tf 6.587 0 Td [(2 j )]TJ/F17 7.9701 Tf 6.587 0 Td [(b m )]TJ/F17 7.9701 Tf 6.586 0 Td [(j + k suchpathswith j p )]TJ/F11 9.9626 Tf 10.308 0 Td [(b )]TJ/F11 9.9626 Tf 10.308 0 Td [(k .Wecompletebysummingoverthe appropriate j Toreversethemapping,itsucestoshowthat 1 2 ,and 3 canbeidentied.Let bea realizationwiththefollowingproperties: 1 X n = b + p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m + forthersttimewhen n = n 1 2 X n = )]TJ/F11 9.9626 Tf 7.748 0 Td [(k forsome nn 1 4 X n = b + k + p )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + forsome n n 1 Then,therealizationuptoandincludingthe n th 1 ballis 1 .The+ k "tripthatfollowsfrom thispointis 2 ,whiletheremainderoftherealizationforms 3 .Wecompletetheinversemapby properlyinterleaving 1 and 2 asintheproofofTheorem4.18. Remark .Notethat 2 ,the+ )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k + m )]TJ/F11 9.9626 Tf 10.844 0 Td [(p + "path,isnotrestrictedfrombelow.Therefore, reectionofthissegmentandconcatenationwith 1 insteadof 3 willnotwork. 138

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8.2.2. RuinwiththeZero-PassStrategy. Calculatingtheruinprobabilitiesassociatedwiththezeropassstrategyissimilar,andweshallbebrief.Lemma8.4establishesaone-to-onecorrespondence betweenruinrealizationswiththezero-passstrategyandruinrealizationswiththezero-betstrategy forwhicha+1"tripfollowsthepointofruin,if k> 0or m p .Let F m;p b;k denotethe realizationsfromthe m;p urnforwhichapermanentgainof k precedesthepointofruinwith bank b ,sothat r b m;p ;1 ; 1= m + p p )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k F m;p b;k : Thenwehave: Theorem8.8. Suppose m p or k> 0 ,andlet k = k +max f 0 ;p )]TJ/F11 9.9626 Tf 9.963 0 Td [(m g .Then F m;p b;k = p )]TJ/F10 6.9738 Tf 6.227 0 Td [(b )]TJ/F10 6.9738 Tf 6.227 0 Td [(k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X j = k m + p )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 j )]TJ/F11 9.9626 Tf 9.963 0 Td [(b m )]TJ/F11 9.9626 Tf 9.962 0 Td [(j + k +1 )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(S b;k +1 j + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( k S b;k j + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 where k =0 if k =0 andequals 1 otherwise. Remark .IfweprovedTheorem8.8inthemanneroftheproofofTheorem8.7,then 2 wouldbea + k +1+ m )]TJ/F11 9.9626 Tf 10.278 0 Td [(p + "trip,withtheextra+1"tripassociatedwiththersttripfromneutralto one. Forthecase m


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8.3. TheRuinProblemfortheRandomAcceptanceUrn. Theruinproblemcansimilarly beappliedtotheBayesianversionoftheurn.Weagaingivetheplayerabank b = B 0 ,andmonitor therandomvariables B i ,0 i n .Sincetheoptimalstrategyisverydependentontheinitial priordistribution ,weshallconcentratesolelyonthe n; ;1 ; 1urns. 8.3.1. TheRuinProbabilitywith s = t =1 Weassumethatourinitialpriordistribution ofthe urnson n ballsmeettheconditionsspeciedbyLemma7.4.Weshallhaveourplayerusethebetting ruleof[7] )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(bettingifandonlyif P k j =1 Y j 0 ,whichweshallagaincallthe zero-betstrategy .Then theplayershallberuinedifandonlyiftheevent .8 Z := k X j =1 Y j =0,thena )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ballisdrawn occursforatleast b valuesof k .Then,theprobabilitythattheplayerisruinedusingthezero-bet strategyequals P Z b .Thisisbecausethepermanentgaintheinitialweightoftheurn,ifitis positiveismadeduringthenaltripfromtheinitialweighttozero.Previoustripsareundone,as theplayerbetsnotonlyfromtheinitialweighttozero,butalsofromzerobacktotheinitialweight, losingwhathadbeengained.Thisdoesnothappenforthelasttrip,astheurnweightnishesat zero. Theorem8.10. Suppose satisestheconditionsofLemma7.4andthezero-betstrategyisused. Thenforanyinteger b 1 ,theprobabilityofruinwithbank b equals X j
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Case2 .If2 js= s + t 141

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isalsoanoptimalstrategy.ForthedistributionssatisfyingtheconditionsofLemma7.4,thisbetting rulecanbestatedwhen s = t =1asbetifandonlyif P k j =1 Y j > 0."Weshallcallthispolicythe zero-passstrategy .Thistime,theplayerisruinediftheevent .9 Z 0 := k X j =1 Y j =1 ; thena )]TJ/F8 9.9626 Tf 7.749 0 Td [(1"ballisdrawn happensatleast b times.Then,theprobabilitythattheplayerisruinedwhileusingthezero-pass strategyequals P Z 0 b Theorem8.11. Suppose satisestheconditionsofLemma7.4.Thenforanyinteger b 1 ,the probabilityofruinusingthezero-passstrategyequals )]TJ/F11 9.9626 Tf 7.749 0 Td [(q n= 2 n n= 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n n= 2+ b +2 X j n= 2 q j n j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n j + b : Proof. WeproceedsimilarlytotheproofofTheorem8.10.Thistime,ifthereare j +1"balls initially,.9occurseachtimetheurnweightpassesfrom2 j )]TJ/F11 9.9626 Tf 10.065 0 Td [(n )]TJ/F8 9.9626 Tf 10.065 0 Td [(1to X 0 j =2 j )]TJ/F11 9.9626 Tf 10.065 0 Td [(n .Given with Z b R hasatleast b passagesfrom0to1.Therefore, R gainsatleastmax f 0 ; 2 j )]TJ/F11 9.9626 Tf 8.529 0 Td [(n g + b fromthenonrandom n )]TJ/F11 9.9626 Tf 9.963 0 Td [(j;j ;1 ; 1urn.Therefore, P Z 0 b j j = P )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(playergainsatleastmax f 0 ; 2 j )]TJ/F11 9.9626 Tf 9.963 0 Td [(n g fromthe n )]TJ/F11 9.9626 Tf 9.962 0 Td [(j;j ;1 ; 1urn = 8 > > < > > : n j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 n j + b ; if2 j
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UsingMaple12 TM ,wehavecalculatedsomeruinprobabilitiesusingthezero-passstrategy,and presentthemwithTable8.4forcomparisonwiththevaluesofTable8.3. Table8.4. Probabilitiesofruinapproximatedtothreedecimalplaces,with uniform,usingthezero-passstrategy. bank n =5 n =10 n =100 n =1000 n =10000 b =1.233.306.377.385.386 b =2.033.101.209.225.227 b =30.026.132.156.159 b =40.004.088.118.121 b =50.000.059.093.098 b =600.040.076.082 b =700.026.064.070 b =800.017.055.061 143

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9. Conclusion Wehavefoundthatacceptanceurnmodelcanbegeneralizedtotheurnscontainingballsofvalues )]TJ/F11 9.9626 Tf 7.749 0 Td [(s and+ t .Liketheoriginalacceptanceurnmodelof[7],anoptimalstrategythatmaximizesthe expectedgainistobetwhenevertheurnweightispositive )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(thatis,whentheprobabilitythenext ballhasvalue+ t isatleast s= t + s ,andtopasswhenevertheurnweightisnegative.Itwould notbeasurprise,then,thatthiswouldalsobethecasewhentheurncontainsballsofmultiple positiveandnegativevalues.Moreover,themethodofcountingcrossingscanbeappliedinthiscase, thoughanyformulaswillbeextremelycomplicated.Itisalsolikelythattheproblemofndingthe probabilitytheplayergainsatleast k isequivalenttondingtheprobabilitythatthemaximum weightoftheurnequalsisatleast k .Inthiswaytheacceptanceurnmodelcanbeinterpreted asanextensionoftheballotproblem,withthequestionWhatistheprobabilitythatcandidate A nevertrailsbymorethan k inweightasthevotesarecounted?". Theuseofverticalstretchingisnatural,andprovedtobeveryusefulforscalingpurposes. However,itonlyhasaglobaluseforthe m;p ; s;t urnsLemma2.1,andcannotbeusedlocally, duetotherestrictionsplacedontheballvalues.Verticalreectionisalsoanaturaltooltouse,and gaveaglobalresult,theAntiurnTheorem.Thereectionmethod,fortheseacceptanceurns,hasa localapplication,butonlyinthecase s = t .Fortheurnswith s = t =1,wecanmaprealizations fromthe m;p ;1 ; 1urnstorealizationsfromany m 0 ;p 0 ;1 ; 1urncontainingthesamenumberof balls m 0 + p 0 = m + p .Thesemappingsproducedelementaryproofsforboththedistributionofthe gainusinganoptimalstrategyfortheurnLemma3.4,andthemaximumweightproblemLemma 3.5thatdidnotrequireadditionalshuingofthepaths.Theruinproblemwasalsosolvedforthe m;p ;1 ; 1urnswithbank b Theorems8.7,8.8,and8.9,usinginclusion/exclusionandrepeated reection,thoughsomeshuingwasrequiredfortheproofsofthoseresults.Unfortunately,localized reectioncannotbeusedingeneralforthe m;p ; s;t urns,andasaresultsolvingtheseproblems becamemoredicultandresultsweremorecomplicated. Rotation,unlikeverticalstretchingandreection,isauniversaltransformation,andinparticular itcanbeappliedlocallyforallofthe m;p ; s;t urns.However,itsmainlimitationisthatarotation 144

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doesnotchangethecompositionofthepathatall,as and i;j ]arebothrealizationsfrom thesameurn.Asaresult,theuseofrotationgloballyisoflimiteduse,whentheexpectedgain G m;p ; s;t isconsidered,astheonlyresultinthisworkthatusesglobalrotationisTheorem4.15. Thereissignicantsymmetrywiththe m;p ; s;t urns,asshownwiththeregularrepeatingcycle" ofthecrossings,butrotationdoesnotseemtobeagoodenoughtoolingeneraltotakeadvantage ofthissymmetry. Therotationalsymmetryfoundforthecase s =1and t apositiveintegerisperhapsthemost noteworthyresultofthiswork.ForthoseurnsandtheonesindicatedbyLemma2.1andthe AntiurnTheorem,rotationturnedouttobeasucienttooltoproperlyusethesymmetry,andwe obtainedafundamentalresult,theCrossingLemmaLemma3.7.TheCrossingLemmachanged theproblemofcountingcrossingstothesimplerproblemofcountingthenumberoftimestheurn isneutral,for m pt .Countingthenumberoftimestheurnwasneutralcouldbedoneinmany dierentways,alsoasaresultoftheCrossingLemma.Thisinducedcombinatorialidentities,which couldbeshowninalargercontextusinggeneralizedbinomialseriesLemma3.1.Thecounting methodsusedfortheproofsofLemmas3.10and3.13canalsobeusedtocountsimilarobjectsat anyxedweightoftheurn,andthustheCrossingLemmahasmoregeneralapplicationsinthearea oflatticepaths.See[23]. When m
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wewishtondhowtoexpress when m and p arelarge.SomecalculationswithMaple12 TM ,along withresultsusingtheCrossingLemmafromthecorrespondingrandomwalkleadustomakethe followingconjectures. Conjecture9.1. Suppose t isapositiveinteger,andthat pt )]TJ/F11 9.9626 Tf 9.45 0 Td [(m> 0isaxedconstant.Thenthe sequence f m;p ;1 ;t g ismonotonicallyincreasing,andconvergesas p !1 Conjecture9.2. Ifas p !1 pt )]TJ/F11 9.9626 Tf 11.062 0 Td [(m !1 and pt )]TJ/F11 9.9626 Tf 11.063 0 Td [(m = o p ,thenfor0 i t )]TJ/F8 9.9626 Tf 11.062 0 Td [(1, P X = )]TJ/F11 9.9626 Tf 7.748 0 Td [(i t )]TJ/F11 9.9626 Tf 10.24 0 Td [(i =T t ,where T t = P t i =1 i = t t +1 = 2isthe t th triangularnumber.Thus,as p !1 wehave m;p ;1 ;t 1 T t t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X i =1 i t )]TJ/F11 9.9626 Tf 9.962 0 Td [(i = t )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 3 : Weexpectthecondition pt )]TJ/F11 9.9626 Tf 9.543 0 Td [(m = o p toberequired,sinceforverylargeurnweightstheexpected driftisfarfromzero.Itisunclearhowtocomputeanapproximate"mappingtoexplainthis asymptoticresult. Thecrossingnumberproblemhasasimilarextensiontothe m;p ; s;t urns,anditssignicanceis thatitwillresultintheplayer'srstnontrivialpermanentgainduringplayonthe m;p ; s;t urn.For t=s rational,theproblemcanbereducedtothecasewith s and t positiveintegerswithgcd s;t =1, and ms )]TJ/F11 9.9626 Tf 9.063 0 Td [(pt 0,inwhichcasetherstzero-betup-crossingisto0 ;:::;s )]TJ/F8 9.9626 Tf 9.063 0 Td [(1.However,becauseof theorderingofthecrossings,wecannotreducetherange m;p ; s;t asdenedinDenition6.23 inthemannerofLemma3.23.Wealsodonothaveaformof G m;p ; s;t thatinvolvesthecrossing number,soweareunabletoproduceevenacatastrophic"cancellationformof m;p ; s;t Forthe m;p ; s;t urns,wewereabletoshowthebasicstructureof G m;p ; s;t inChapter2. However,themethodofurnsimulationforexample,theproofofLemma2.6doesnotappearto workuponconsiderationofthesecond-orderdierences.Let 2 G m ;p ; s;t = G m;p +2; s;t + G m;p ; s;t )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 G m;p +1; s;t ; 2 G p m; ; s;t = G m +2 ;p ; s;t + G m;p ; s;t )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 G m +1 ;p ; s;t ; 2 G m;p ; s;t = G m +2; p ; s;t + G m;p +2; s;t )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 G m +1 ;p +1; s;t : Weconjecturethatthesesecond-orderdierencessharethesamepropertiesasthosewith s = t =1. Conjecture9.3. Forpositive s and t ,andpositiveintegers m and p ,wehave 2 G m ;p ; s;t > 0, 2 G p m; ; s;t > 0,and 2 G m;p ; s;t > 0. 146

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Itshouldbenotedthat 2 G m ;p ; s;t = 2 G m p; ; t;s bytheAntiurnTheorem. MostoftheasymptoticresultsofChapter5wereobtainedusingthepseudo-binomialform.2, which,when s =1, t isapositiveinteger,and m pt ,isveryclosetothebinomialformTheorem 3.15.Unfortunately,if ms= pt > 1,thetransitionof G m;p ; s;t tothepseudo-binomialform introducesanerrortermthatistoolarge,andinsteadweareleftwiththelesssatisfyingresult G m;p ; s;t = O Lemma5.19forthiscase.SomecalculationswithMaple12 TM indicatethat thenaturalcandidatefortheasymptoticvalueof G m;p ; s;t t + s 2 )]TJ/F15 7.9701 Tf 6.587 0 Td [(1 ,doesnotappeartobeequal totheasymptoticvalue.Inparticular,with s =3, t =5,and =2,thenaturalcandidategives4, but G ; 300;3 ; 5 > 4 : 0326.With s =2, t =3,and =2 : 5, G ; 2000;2 ; 3 > 1 : 709 > 5 = 3. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(Wealsobelievethattheexpectedgainwillincreaseas m increases,asshownin[7,Theorem2.7]. Asaresult,wecannotspeculateontheasymptoticvalueof G m;p ; s;t with ms= pt > 1in thiswork. TheBayesianversionoftheacceptanceurnmodelhasabroaderreal-worldapplication,sincethe futurepriceofacommodityisgenerallyunknown.Generally,wehaveshownthatifthemarketis morelikelytogoup,thentheplayershouldbet,andtheplayershouldnotbetifthemarketismore likelytogodown.Foraselectgroupofdistributionswith s = t =1,simplerulesweregivenbased ontheoutcomeofthegamethroughtherst k balls,butanicerdescriptionofabettingruleseems tobediculttoobtainforageneralinitialpriordistribution .Asaresult,ndingtheexpected gainforotherdistributionsisanopenquestionforthe n; ; s;t urns,aswellasthe n; ;1 ;t urns with t apositiveintegerandthe n; ; s;t urns.ThealgorithmsgivenwithTheorems7.10for thecase s = t =1and7.14indicateswhentheplayershouldbet,andoutputstheexpectedgain G n; ; s;t ,forany n and ,butunfortunatelythealgorithmitselfrequiresmanycalculations,even when s = t Wesolvedtheruinprobleminthecase s = t =1forany m p ,withaninitialbank b ,andthe useofanoptimalstrategythatmaximizestheexpectedgain.Unfortunately,theformulaobtained isquitecomplicated.Usingtherathersoftnecessaryconditionforruinandthesimplesolution, wewereonlyabletoobtainalimitednumberofasymptoticresults.Themethodoftheproofof Theorem8.7issoundwhenappliedtothe m;p ;1 ;t urnswith t apositiveinteger.However,wedo nothavethesolutiontoageneralizedversionoftheproblemfoundinFeller[9]Lemma8.6,and certainlythemethodoftheproofofLemma8.6willnolongerapply.Attheveryleast,wecangive adescriptionofwhatarealizationproducingruinlookslikeintermsofthemappingusedinthe 147

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proofofTheorem8.7.Forthe m;p ; s;t urns,theruinproblemisfurthercomplicatedbythefact thattheplayercanberuinedif B n equals0 ;:::;s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1. Wehavefoundanecessaryconditionforruin.Wecanndasucientconditionforruinaswell, whichcanbeexpressedintermsofthegeneralizedballotnumbersofChapter6.Unfortunately,this sucientconditionisquitesoft,andasaresultwecannotprovethattheprobabilityofruinisnear onewhenthebankissmall.However,wedobelievethistobethecase. Conjecture9.4. If pt )]TJ/F11 9.9626 Tf 9.962 0 Td [(ms = O p p and B 0 = o p p ,thenforthe m;p ; s;t urnwehave P B min
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ListofReferences [1]Aeppli,A., ZurTheorieverketteterWahrscheinlichkeiten ,These.Zurich,1924. [2]Andre,D.SolutiondirecteduproblemeresoluparM.Bertrand. ComptesRendusAcad.Sci. Paris ,105,pp.436-437. [3]Bertrand,J.Solutiond'unprobleme," ComptesRendusAcad.Sci.Paris ,105,p.369. [4]Boyce,W.StoppingRulesforSellingBonds, BellJ.Econ.ManageSci. ,1,pp.27-53. [5]Boyce,W.OnaSimpleOptimalStoppingProblem, DiscreteMath. ,5,pp.297-312. [6]Chen,R.,andHwang,F.Onthevaluesofan m;p Urn, Congr.Numer. ,41,pp.75-84. [7]Chen,R.,Zame,A.,Odlyzko,A.,andShepp,L.,AnOptimalAcceptancePolicyforanUrn Scheme, SIAMJ.DiscreteMath. ,11,pp.183-195. [8]Chen,R.,Zame,A.,Lin,C.,andWu,H.ARandomVersionofShepp'sUrnScheme, SIAMJ. DiscreteMath. ,19,pp.149-164. [9]Feller,W., AnIntroductiontoProbabilityTheoryanditsApplications ,Vol.1,3rdedition.New York:Wiley,1968. [10]Graham,R.,Knuth,D.,andPatashnik,O., ConcreteMathematics:AFoundationforComputer Science ,2 nd edition.NewYork:Addison-Wesley,1994. [11]Goulden,I.andSerrano,L.MaintainingtheSpiritoftheReectionPrincipleWhentheBoundaryHasArbitraryIntegerSlope. J.Combin.TheorySer.A ,104,pp.317-326. [12]Irving,J.andRattan,A.TheNumberofLatticePathsBelowaCyclicallyShiftingBoundary, J.Combin.TheorySer.A ,116,pp.499-514. [13]Krattenthaler,C.,Theenumrationoflatticepathswithrespecttotheirnumberofturns,in: N.BalakrishnanEd., AdvancesinCominatorialMethodsandApplicationstoProbabilityand Statistics ,Birkhauuser,Boston,29-58,1997. [14]Mohanty,S.G., LatticePathCountingandApplications .NewYork:AcademicPress,1979. [15]Mohanty,S.G.,OnSomeGeneralizationofaRestrictedRandomWalk, StudiaSci.Math. Hungar. ,3,pp.225-241. [16]Raney,G.N.FunctionalCompositionPatternsandPowerSeriesReversion, Trans.Amer.Math. Soc. ,94,pp.441-451. [17]Ratsaby,J.EstimateoftheNumberofRestrictedInteger-Partitions, Appl.Anal.Discrete Math. ,2,pp.222-233. [18]Renault,M.FourProofsoftheBallotProblem, Math.Mag. ,80,pp.345-352. 149

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[19]Renault,M.LostandFoundinTranslation:Andre'sActualMethodandItsApplicationto theGeneralizedBallotProblem, Amer.Math.Monthly ,115,pp.358-363. [20]Shepp,L.ExplicitSolutionstoSomeProblemsofOptimalStopping, Ann.Math.Statist. ,40 ,pp.993-1010. [21]Suen,S.,andWagner,K.,AnAsymptoticFormulafortheTailofaBinomialDistribution, 2006,unpublished. [22]Suen,S.andWagnerK.,SomeResultsfortheAcceptanceUrnModel,2009,acceptedfor publication,withminorchanges,in SIAMJ.Disc.Math. [23]Suen,S.andWagner,K.,TheEnumerationofSequenceswithRestrictionsonTheirPartial Sums,submitted. [24]Takacs,L.,AGeneralizationoftheBallotProblemandItsApplicationintheTheoryofQueues, J.Amer.Statist.Assoc. ,57,pp.327-337. 150

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AbouttheAuthor KevinP.WagnerreceivedbothaBachelor'sandMaster'sDegreeinMathematicsin2001from theUniversityofSouthFloridaasapartofthe5-YearB.A./M.A.Program.Heservedasateaching assistantintheMathematicsDepartmentattheUniversityofSouthFloridafrom2000to2006,and wasaregularattendeeattheDiscreteMathematicsSeminarduringthattime.Hehascoauthored apapertobepublishedinthe SIAMJournalonDiscreteMathematics inlate2010orearly2011.