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Some problems on products of random matrices

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Some problems on products of random matrices
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Cureg, Edgardo S
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University of South Florida
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Tampa, Fla
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Weak convergence
Stochastic matrices
Circulant matrices
Lyapunov exponents
Random Fibonacci sequences
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
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theses   ( marcgt )
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Abstract:
ABSTRACT: We consider three problems in this dissertation, all under the unifying theme of random matrix products. The first and second problems are concerned with weak convergence in stochastic matrices and circulant matrices, respectively, and the third is concerned with the numerical calculation of the Lyapunov exponent associated with some random Fibonacci sequences. Stochastic matrices are nonnegative matrices whose row sums are all equal to 1. They are most commonly encountered as transition matrices of Markov chains. Circulant matrices, on the other hand, are matrices where each row after the first is just the previous row cyclically shifted to the right by one position.^ ^Like stochastic matrices, circulant matrices are ubiquitous in the literature.In the first problem, we study the weak convergence of the convolution sequence mu to the n, where mu is a probability measure with support S sub mu inside the space S of d by d stochastic matrices, d greater than or equal to 3. Note that mu to the n is precisely the distribution of the product X sub 1 times X sub 2 times and so on times X sub n of the mu distributed independent random variables X sub 1, X sub 2, and so on, X sub n taking values in S. In CR Santanu Chakraborty and B.V. Rao introduced a cyclicity condition on S sub mu and showed that this condition is necessary and sufficient for mu to the n to not converge weakly when d is equal to 3 and the minimal rank r of the matrices in the closed semigroup S generated by S sub mu is 2. Here, we extend this result to any d bigger than 3.^ ^Moreover, we show that when the minimal rank r is not 2, this result does not always hold.The second problem is an investigation of weak convergence in another direction, namely the case when the probability measure mu's support S sub mu consists of d by d circulant matrices, d greater than or equal to 3, which are not necessarily nonnegative. The resulting semigroup S generated by S sub mu now lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence mu to the n to analyze the problem.^ Our approach is based on the work of Mukherjea and his collaborators, who in LM and DM presented a method based on a bookkeeping of the possible structure of the compact kernel K of S.The third problem considered in this dissertation is the numerical determination of Lyapunov exponents of some random Fibonacci sequences, which are stochastic versions of the classical Fibonacci sequence f sub (n plus 1) equals f sub n plus f sub (n minus 1), n greater than or equal to 1, and f sub 0 equal f sub 1 equals 1, obtained by randomizing one or both signs on the right side of the defining equation and or adding a "growth parameter." These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent.^ ^Following techniques presented by Embree and Trefethen in their numerical paper ET, we study the behavior of the Lyapunov exponents as a function of the probability p of choosing plus in the sign randomization.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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by Edgardo S. Cureg.
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Title from PDF of title page.
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Document formatted into pages; contains 105 pages.
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Includes vita.

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ABSTRACT: We consider three problems in this dissertation, all under the unifying theme of random matrix products. The first and second problems are concerned with weak convergence in stochastic matrices and circulant matrices, respectively, and the third is concerned with the numerical calculation of the Lyapunov exponent associated with some random Fibonacci sequences. Stochastic matrices are nonnegative matrices whose row sums are all equal to 1. They are most commonly encountered as transition matrices of Markov chains. Circulant matrices, on the other hand, are matrices where each row after the first is just the previous row cyclically shifted to the right by one position.^ ^Like stochastic matrices, circulant matrices are ubiquitous in the literature.In the first problem, we study the weak convergence of the convolution sequence mu to the n, where mu is a probability measure with support S sub mu inside the space S of d by d stochastic matrices, d greater than or equal to 3. Note that mu to the n is precisely the distribution of the product X sub 1 times X sub 2 times and so on times X sub n of the mu distributed independent random variables X sub 1, X sub 2, and so on, X sub n taking values in S. In [CR] Santanu Chakraborty and B.V. Rao introduced a cyclicity condition on S sub mu and showed that this condition is necessary and sufficient for mu to the n to not converge weakly when d is equal to 3 and the minimal rank r of the matrices in the closed semigroup S generated by S sub mu is 2. Here, we extend this result to any d bigger than 3.^ ^Moreover, we show that when the minimal rank r is not 2, this result does not always hold.The second problem is an investigation of weak convergence in another direction, namely the case when the probability measure mu's support S sub mu consists of d by d circulant matrices, d greater than or equal to 3, which are not necessarily nonnegative. The resulting semigroup S generated by S sub mu now lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence mu to the n to analyze the problem.^ Our approach is based on the work of Mukherjea and his collaborators, who in [LM] and [DM] presented a method based on a bookkeeping of the possible structure of the compact kernel K of S.The third problem considered in this dissertation is the numerical determination of Lyapunov exponents of some random Fibonacci sequences, which are stochastic versions of the classical Fibonacci sequence f sub (n plus 1) equals f sub n plus f sub (n minus 1), n greater than or equal to 1, and f sub 0 equal f sub 1 equals 1, obtained by randomizing one or both signs on the right side of the defining equation and or adding a "growth parameter." These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent.^ ^Following techniques presented by Embree and Trefethen in their numerical paper [ET], we study the behavior of the Lyapunov exponents as a function of the probability p of choosing plus in the sign randomization.
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Weak convergence.
Stochastic matrices.
Circulant matrices.
Lyapunov exponents.
Random Fibonacci sequences.
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PAGE 1

Some Problems on Products of Random Matrices by Edg ardo S. Cure g A dissertation submitted in partial fulllment of the requirements for the de gree of Do ctorate of Philosop y Department of Mathematics Colle ge of Arts and Sciences Univ ersit y of South Florida Major Professor: Aruna v a Mukherjea, Ph.D. A thanassios Kartsatos, Ph.D. Jogindar Ratti, Ph.D. Stephen Suen, Ph.D Date of Appro v al: No v em b er 14, 2006 Keyw ords: w eak con v ergence, sto c hastic matrices, circulan t matrices, Ly apuno v exp onen ts, random Fib onacci sequences c Cop yright 2006 Edg ardo S. Cure g

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Dedication T o the memory of my late f ather Bernardo C. Cure g, Sr (1931-1991), who taught me the v alue of education.

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Ackno wledgments This w ork o wes much of its de v elopment to the helpful support, guidance, suggestions, ideas, criticism, and patience of the author' s major professor Dr Aruna v a Mukherjea, whom he sincerely thanks for e v erything concerned with the inception, production, and completion of this dissertation. It has been more than v e years after Dr Mukherjea g a v e a course in random matrices at USF in the summer of 2001, thus introducing the author to this eld, and since then it has been an incredible learning e xperience – no, journe y – for him. Thank you so much, Dr Mukherjea! The author w ould also lik e to thank his committee members, Drs. Dennis Killinger Athanassios Kartsatos, Jogindar Ratti, and Stephen Suen, for re vie wing and e v aluating this dissertation. He also e xtends his sincerest gratitude to A yak o Osso wski, Nanc y Morris, and Sarina Maldonado, for all the help with the paperw ork and documentation in v olv ed with this dissertation; and to Dr Da vid K ephart, one of his most trusted friends and whose mathematical acuity he so admires and en vies, not only for pro viding the L A T E X template for this dissertation, b ut for his f aithful friendship since the v ery be ginning. The author wishes to ackno wledge the assistance pro vided by the National Science F oundation through their Graduate Research Fello wship Program. NSF has pro vided the author not only with funds for tuition and stipend from Summer 2001 to Summer 2004, b ut, more importantly the opportunity to conduct collaborati v e w ork with international researchers in T aiw an and Finland. He e xtends his deepest appreciation to Prof. G ¨ oran H ¨ ogn ¨ as of Abo Akademi Uni v ersity in T urku, Finland, for hosting him as an NSF international research tra v el grantee during the summer of 2004, and for in viting him to the 27th Finnish Summer School on Probability He also thanks Prof. Y eong-Nan Y eh of the Institute of Mathematics, Academia Sinica, T aipei, T aiw an, for hosting him as a participant to the 2002 NSF Summer Programs in Japan, K orea, and T aiw an (no w called NSF East Asia and P acic Summer Institutes for U.S. Graduate Students (EAPSI)). Thanks are also due Prof. Nata sa Jonoska of USF' s Department of Mathematics, as well as Profs. Arlene P ascasio, Blessilda Raposa, and Leonor Rui vi v ar of the Mathematics Department, De La Salle Uni v ersity Manila, Philippines, for their support of the author' s application for the NSF Graduate Research Fello wship. The author w ould also lik e to thank Japan' s Ministry of Education, Culture, Sports, Science and T echnology (formerly kno wn as Monb usho ) for the scholarships it pro vided from 1988 to 1992 at K umamoto National Colle ge of T echnology and from 1992 to 1994 at T o yohashi Uni v ersity of T echnology and the T ok yo-based IBM Asia for a 1994-1996 fello wship at TUT He of fers his deepest thanks to his Japanese friend, T akik o “T ina” Nag aya, for her lo v e and generosity through all these years, and to his sister Leila T aghap, whom he lo v es dearly and to whom he is fore v er indebted, for her nancial support of his early education in the Philippines. Finally the author w ould lik e to thank his best friend, Dr Harry Bro wn, for the moral support during times it w as most needed. Y ou kno w the number Harry!

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T able of Contents List of Figures . . . . . . . . . . . . . . . . . . . ii Abstract . . . . . . . . . . . . . . . . . . . . iii Chapter 1 Introduction . . . . . . . . . . . . . . . . 1 Chapter 2 W eak Con v er gence in Stochastic Matrices . . . . . . . . 8 2.1 Introduction . . . . . . . . . . . . . . . . . 8 2.2 Preliminaries . . . . . . . . . . . . . . . . 10 2.3 4 £ 4 Stochastic Matrices . . . . . . . . . . . . . 21 2.4 The d £ d case . . . . . . . . . . . . . . . . 36 Chapter 3 W eak Con v er gence in Circulant Matrices . . . . . . . . 49 3.1 d £ d Circulant Matrices . . . . . . . . . . . . . 49 3.2 3 £ 3 Circulant Matrices . . . . . . . . . . . . . 56 3.3 4 £ 4 Circulant Matrices . . . . . . . . . . . . . 66 3.4 d £ d T oeplitz Matrices . . . . . . . . . . . . . . 76 Chapter 4 Numerical Calculation of L yapuno v Exponents for Some Random Fibonacci Recurrences . . . . . . . . . . 79 4.1 Random Matrix Products and Random Fibonacci Sequences . . . . 79 4.2 Numerical Calculation of the L yapuno v Exponents . . . . . . 85 References . . . . . . . . . . . . . . . . . . . . 104 About the Author . . . . . . . . . . . . . . . . End P age i

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List of Figures 1 1 ( p ) for dif ferent v alues of p . . . . . . . . . . . . . 88 2 2 ( p ) for dif ferent v alues of p . . . . . . . . . . . . . 89 3 ( p ) ; where = 1 = 2 ; for dif ferent v alues of p . . . . . . . . 90 4 ( p ) ; where = 3 = 4 ; for dif ferent v alues of p . . . . . . . . 91 5 ( p ) ; where = 2 ; for dif ferent v alues of p . . . . . . . . . 92 6 ( p ) ; where = 8 ; for dif ferent v alues of p . . . . . . . . . 93 7 ( p ) for x ed p = 0 : 2 and dif ferent v alues of < 1 . . . . . . . 94 8 ( p ) for x ed p = 0 : 4 and dif ferent v alues of < 1 . . . . . . . 95 9 ( p ) for x ed p = 0 : 6 and dif ferent v alues of < 1 . . . . . . . 96 10 ( p ) for x ed p = 0 : 8 and dif ferent v alues of < 1 . . . . . . . 97 11 ( p ) for x ed p = 0 : 2 and dif ferent v alues of > 1 . . . . . . . 98 12 ( p ) for x ed p = 0 : 4 and dif ferent v alues of > 1 . . . . . . . 99 13 ( p ) for x ed p = 0 : 6 and dif ferent v alues of > 1 . . . . . . . 100 14 ( p ) for x ed p = 0 : 8 and dif ferent v alues of > 1 . . . . . . . 101 15 1 vs. p . . . . . . . . . . . . . . . . . . . 102 16 2 vs. p . . . . . . . . . . . . . . . . . . . 102 17 vs. p for = 0 : 5 ; 0 : 75 ; 2 ; and 8 . . . . . . . . . . . . 103 ii

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Some Problems on Products of Random Matrices Edg ardo S. Cure g ABSTRA CT W e consider three problems in this dissertation, all under the unifying theme of random matrix products. The rst and second problems are concerned with weak con v er gence in stochastic matrices and circulant matrices, respecti v ely and the third is concerned with the numerical calculation of the L yapuno v e xponent associated with some random Fibonacci sequences. Stochastic matrices are nonne g ati v e matrices whose ro w sums are all equal to 1. The y are most commonly encountered as transition matrices of Mark o v chains. Circulant matrices, on the other hand, are matrices where each ro w after the rst is just the pre vious ro w c yclically shifted to the right by one position. Lik e stochastic matrices, circulant matrices are ubiquitous in the literature. In the rst problem, we study the weak con v er gence of the con v olution sequence n ; where is a probability measure with support S inside the space S of d £ d stochastic matrices, d 3 : Note that n is precisely the distrib ution of the product X 1 X 2 ¢ ¢ ¢ X n of the -distrib uted independent random v ariables X 1 ; X 2 ; : : : ; X n taking v alues in S : In [CR ] Santanu Chakraborty and B.V Rao introduced a c yclicity condition on S and sho wed that this condition is necessary and suf cient for n to not con v er ge weakly when d = 3 and the minimal rank r of the matrices in the closed semigroup S generated by S is 2 : Here, we e xtend this result to an y d > 3 : Moreo v er we sho w that when the minimal rank r is not 2 ; this result does not al w ays hold. iii

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The second problem is an in v estig ation of weak con v er gence in another direction, namely the case when the probability measure s support S consists of d £ d circulant matrices, d 3 ; which are not necessarily nonne g ati v e. The resulting semigroup S generated by S no w lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence n to analyze the problem. Our approach is based on the w ork of Mukherjea and his collaborators, who in [LM] and [DM] presented a method based on a bookk eeping of the possible structure of the compact k ernel K of S : The third problem considered in this dissertation is the numerical determination of L yapuno v e xponents of some random Fibonacci sequences, which are stochastic v ersions of the classical Fibonacci sequence f n +1 = f n + f n ¡ 1 ; n 1 ; and f 0 = f 1 = 1 ; obtained by randomizing one or both signs on the right side of the dening equation and/or adding a “gro wth parameter ” These sequences may be vie wed as coming from a sequence of products of i.i.d. random matrices and their rate of gro wth measured by the associated L yapuno v e xponent. F ollo wing techniques presented by Embree and T refethen in their numerical paper [ET ], we study the beha vior of the L yapuno v e xponents as a function of the probability p of choosing + in the sign randomization. i v

PAGE 8

Chapter 1 Introduction T o set the frame w ork for the problems we consider in this dissertation, we recall the denition of weak con v er gence in the conte xt of measures on topological semigroups [HMu ]. Let S be a locally compact Hausdorf f second-countable topological semigroup. Let B be the class of Borel subsets of S : Let P ( S ) be the set of all re gular probability measures dened on B ; i.e. measures that satisfy the condition that for e v ery > 0 ; there e xists a compact set K 2 B for which ( S n K ) < : The support S of 2 P ( S ) is gi v en by S = f x 2 S : ( V ) > 0 for an y open set V containing x g : Note that S is closed and ( S ) = 1 : A sequence ( n ) n 1 in P ( S ) is then said to be weakly con v er gent to 2 P ( S ) if lim n !1 Z f d n = Z f d for e v ery bounded (real) continuous function f on S : If 2 P ( S ) and X 1 ; X 2 ; : : : ; X n are -distrib uted independent random v ariables taking v alues in S ; then the product X 1 X 2 ¢ ¢ ¢ X n has distrib ution ¤ ¤ ¢ ¢ ¢ ¤ ( n f actors ) n : Here, the con v olution product ¤ of ; 2 P ( S ) is the unique re gular probability measure on B guaranteed by the Riesz representation theorem to e xist and satisfy the equation Z f d ( ¤ ) = Z Z f ( xy ) ( dx ) ( dy ) 1

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for e v ery bounded countinuous function f on S with compact support. Or more con v eniently ¤ ( B ) = Z ( B x ¡ 1 ) ( dx ) = Z ( x ¡ 1 B ) ( dx ) (1. 1) for B 2 B : The sets B x ¡ 1 and x ¡ 1 B in (1. 1) are dened by B x ¡ 1 f y 2 S : y x 2 B g and x ¡ 1 B f y 2 S : xy 2 B g ; respecti v ely The rst tw o problems considered here may then be simply described as the determination of necessary and suf cient conditions for weak con v er gence of the con v olution sequence ( n ) n 1 ; when has support S consisting of either d £ d stochastic matrices or d £ d circulant matrices, and S = [ n 1 S n is the closed (multiplicati v e) semigroup generated by S : Stochastic matrices are nonne g ati v e matrices whose ro w sums are all equal to one. Ubiquitous in the literature, the y are most commonly encountered as transition matrices of Mark o v chains. The moti v ation for studying weak con v er gence in stochastic matrices comes from the w ork of Mukherjea and his students. In [LM] and [DM], Lo and Mukherjea and Dhar and Mukherjea present solutions to the problem of weak con v er gence of con v olution po wers n of a probability measure with support S in d £ d nonne g ati v e and stochastic matrices, respecti v ely in terms of easily v eriable conditions on S : As Dhar and Mukherjea note in their paper the problem of con v er gence in distrib ution of products of d £ d i.i.d. stochastic matrices is an old one, reaching as f ar back as Rosenblatt' s 1965 w ork [MR2]. A method that e xtends to d = 3 Mukherjea' s simple and complete result for d = 2 that weak 2

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con v er gence occurs if and only if the support of is not equal to 8<: 0@ 0 1 1 0 1A 9=; [M1], is presented in their paper This method is based on a general result gi ving a necessary and suf cient condition for weak con v er gence of a tight sequence ( n ) of con v olution po wers of a probability measure (that is, gi v en > 0 ; there e xists a compact subset K of S such that n ( K ) > 1 ¡ for all n 1 ) and looks at the possible structure of the compact k ernel K (which, for semigroups of stochastic matrices, is well-kno wn to consist of matrices with the minimal rank [HMu]) of the closed semigroup S generated by S : The method “w orks for d > 3 e v en though calculations are more in v olv ed for higher v alues of d ” In another paper dealing with the same subject for d = 3 ; Chakraborty and Rao [CR ] present a dif ferent solution, this time based on a di vision of S into certain subsets according to the number of recurrent and transient classes. Their method is “too cumbersome to be carried o v er to higher dimensions, ” b ut it succeeded in e xpressing the same result obtained by Dhar and Mukherjea in more succint terms using a c yclicity property of S : According to their denition, S is c yclic if there are pairwise disjoint subsets A 1 ; A 2 ; : : : ; A k of f 1 ; 2 ; : : : ; d g such that for an y s; 1 s k ; and for an y i 2 A s ; X j 2 A s +1 x ij = 1 for an y element x = ( x ij ) 2 S : Note that A k +1 = A 1 in the sum. Chakraborty and Rao then pro v ed in [CR] that when d = 3 ; as long as S is not contained in a group of permutation matrices, n does not con v er ge weakly if and only if S is c yclic. The e xpansion of the methods presented in [DM] and [CR] to 4 £ 4 stochastic matrices, as well as the generalization of Chakraborty and Rao' s result in the case when the rank of the matrices in the k ernel K of the semigroup S generated by S ; where S is contained in a set of d £ d stochastic matrices, d 4 ; is 2 ; is essentially the subject of our rst problem. P art of the ndings in this in v estig ation is that in the general d £ d situation, this 3

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characterization of non-weak con v er gence of n in terms of c yclicity of S is no longer v alid when the rank of the matrices in the k ernel K of S is bigger than 2 : In obtaining our main results, we follo wed Dhar and Mukherjea' s approach and used the general theorem mentioned abo v e. From the proofs presented here, ho we v er it is e vident that the connection between c yclicity and non-weak con v er gence does not follo w easily from this general result. Lea ving the domain of stochastic matrices and their nonne g ati vity property we ne xt in v estig ate the problem of weak con v er gence of n in the conte xt of circulant matrices, which are not necessarily nonne g ati v e. Circulant matrices are matrices where each ro w after the rst is just the pre vious ro w c yclically shifted to the right by one position. F amiliar e xamples are the identity matrix, the zero matrix, and the all 1 s matrix. A 3 £ 3 circulant matrix has the form 0BBB@ a b c c a b b c a 1CCCA : These matrices appear in man y mathematical problems. A detailed account of this (with man y e xamples) can be found in the beautiful w ork of Diaconis[Di ]. T ak e, for e xample, Diaconis' Example 2 : 1 (p. 40, [Di ]). He considers “a particle constrained to hop about on n points arranged in a circle. At each time the particle hops left or right with probability 1 2 : This is the c yclic v ersion of the classical drunkard' s w alk. Inde x the points as 0 ; 1 ; 2 ; : : : ; n ¡ 1 : The chance of mo ving from i to j is thus M ij = 8<: 1 2 ; if j i ¡ j j = 1; 0 ; otherwise. The matrix M is a circulant matrix with rst ro w (0 ; 1 2 ; 0 ; : : : ; 0 ; 1 2 ) : ” Diaconis goes on in his paper discussing problems in v olving co v ariance matrices, c yclic codes, etc., where one encounters circulant matrices. Circulant matrices ha v e a nice structure, and as f ar as we kno w the problem of con v er gence in distrib ution of products of d £ d i.i.d. random circulant matrices, or equi v alently 4

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the problem of weak con v er gence of con v olution po wers n of a probability measure supported on such matrices, has not been studied so f ar Hence this is the second problem we consider W e clarify that in the circulant case, S ; the closed semigroup generated by the support S of ; usually is not compact (in the usual topology), so we assume tightness of the sequence ( n ) n 1 : Our main result points to the importance of the special orthogonal group S O ( d ) in the characterization of weak con v er gence of ( n ) : The third problem considered in this dissertation is the numerical determination of L yapuno v e xponents of some random Fibonacci sequences. A random Fibonacci sequence is a stochastic v ersion of the classical Fibonacci sequence f n +1 = f n + f n ¡ 1 ; n 1 ; and f 0 = f 1 = 1 obtained by randomizing one or both of the signs on the right side of the dening equation. F or e xample, one v ersion of a random Fibonacci sequence is the one originally considered by V isw anath in [V i ], gi v en by x n +1 = § x n § x n ¡ 1 ( n 1) (1. 2) with x 0 = x 1 = 1 ; and where the signs are chosen independently and with equal probabilities. V isw anath determined the rate of gro wth of this random sequence. Recall that the rate of gro wth of a random sequence coming from a sequence of i.i.d. random matrices is the e xponential of its associated L yapuno v e xponent, which, by a result of Furstenber g and K esten [FK], is equal to the almost sure limit lim n !1 log j x n j n : In [V i ], V isw anath found the e xact v alue of the rate of gro wth f of the random Fibonacci recurrence (1. 2) to be lim n !1 j x n j 1 =n = e f = 1 : 13198824 : : : (1. 3) with probability 1 : This result w as obtained using “the theory of random matrix products, Stern-Brocot di vision of the real line, a fractal measure, and a rounding error analysis to v alidate the computer calculation. ” Observ e that the rate of gro wth of the classical Fibonacci sequence is gi v en by the golden ratio, 1+ p 5 2 1 : 618 : 5

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V isw anath actually used the random recurrence x n +1 = § x n + x n ¡ 1 (1. 4) in his calculation, since under the same initial conditions and independence and equal probability of choosing the signs, (1. 2) and (1. 4) gi v e rise to the same quantity gi v en in (1. 3). In f act, the recurrence x n +1 = x n § x n ¡ 1 (1. 5) could ha v e been used as well. In his concluding remarks on the subject, V isw anath posed a generalization of the problem in which each § sign is still chosen independently in (1. 2) b ut + and ¡ occur with probabilities p and q := 1 ¡ p; respecti v ely where 0 < p < 1 : Noting that the techniques he used to calculate the L yapuno v e xponent f ( p ) for p = 1 = 2 “do not seem to generalize easily” to arbitrary v alues of p; V isw anath instead calculated f ( p ) numerically for dif fer ent v alues of p using Ulam' s method [HMi ]. The resulting graph of f ( p ) vs. p sho ws a smooth dependence of f ( p ) on p; a result consistent with Peres' theorem [Pe ]. W e emphasize that V isw anath' s numerical calculation of f ( p ) w as done for the random recurrence (1. 2). V isw anath did not consider the numerical approximation of L yapuno v e xponents for the corresponding generalization to the random recurrences (1. 4) and (1. 5). Thus, here we in v estig ate this problem. Our numerical results in this case suggest that the L yapuno v e xponent for (1. 5) e xhibits symmetry with respect to p = 1 = 2 ; whereas for (1. 4) the L yapuno v e xponent monotonically increases with p; b ut not in the same manner as the L yapuno v e xponent for (1. 2) reported by V isw anath. In a related article [ET], Embree and T refethen g a v e a numerical description of what the y called the “L yapuno v constant” ( ) = lim n !1 j x n j 1 =n (with probability 1) for the random Fibonacci recurrence x n +1 = x n § x n ¡ 1 ; (1. 6) where > 0 ; the signs are chosen independently and with equal probabilities, and x 0 = x 1 = 1 : The y found that for a certain range of v alues of the parameter ; the L yapuno v 6

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constant is less than 1 (resulting in the e xponential decay of the solutions to the random recurrence), and for v alues of outside this range, the L yapuno v constant is greater than 1 (hence the solutions gro w e xponentially). The y further observ ed that depends on in a non-smooth, fractal w ay In the section on “Discussions and Generalizations” of the same paper Embree and T refethen posed as one of the modications of the random Fibonacci recurrence (1. 6) the follo wing generalization: “The coin might be weighted, so that + is chosen with probability p and ¡ with probability 1 ¡ p: ” The other problem we consider here is e xactly this generalization. Our results suggest that for v alues of < 1 ; there appears to e xist some p ¤ = p ¤ ( ) for which the L yapuno v e xponent for (1. 6) is 0 ; meaning the random Fibonacci sequence in this case neither gro ws nor decays. 7

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Chapter 2 W eak Con v er gence in Stochastic Matrices 2.1 Introduction Stochastic matrices are nonne g ati v e matrices whose ro w sums are all equal to 1. Ubiquitous in the literature, these matrices are most commonly encountered as transition matrices of Mark o v chains. W e no w describe briey the problem of weak con v er gence in stochastic matrices. Let be a probability measure on B ; the Borel sets of d £ d stochastic matrices. Here, d is a positi v e inte ger greater than 1 Let S be the support of and let S = [ n 1 S n be the closed multiplicati v e semigroup generated by S : Notice that S is then a compact Hausdorf f topological semigroup (with respect to usual matrix topology and matrix multiplication). W e dene the con v olution iterates n in the usual manner In other w ords, for an y B 2 B ; n +1 ( B ) = Z n f y : y x 2 B g ( dx ) ; for all n 1 : W e then say that the sequence ( n ) n 1 weakly con v er ges to a probability measure if and only if lim n !1 Z f d n = Z f d for e v ery bounded (real) continuous function f on S : In [DM], Dhar and Mukherjea present a solution to the problem of weak con v er gence of n when the support S of is contained in a set of 3 £ 3 stochastic matrices. Their solution 8

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is e xpressed in terms of easily v eriable conditions on S ; similar in spirit to Mukherjea' s simple and beautiful result that when S is contained in 2 £ 2 stochastic matrices, weak con v er gence occurs if and only if S 6 = 8<: 0@ 0 1 1 0 1A 9=; [M1]. Their method is based on a bookk eeping of the possible structure of the compact k ernel K of S which, for semigroups of stochastic matrices, is well-kno wn to consist of all matrices in S with the minimal rank [HMu]. In another paper dealing with the same problem, Chakraborty and Rao [CR] introduce a “c yclicity” property for the support S ; calling S cyclic if there are pairwise disjoint subsets A 1 ; A 2 ; : : : ; A k of f 1 ; 2 ; : : : ; d g such that for an y s; 1 s k ; and for an y i 2 A s ; X j 2 A s +1 P ij = 1 for an y element P = ( P ij ) 2 S : Note that A k +1 = A 1 in the sum. F or e xample, the set 8>>><>>>: 0BBB@ 1 0 0 0 0 1 0 1 0 1CCCA ; 0BBB@ 1 ¡ a ¡ b a b 0 0 1 0 1 0 1CCCA 9>>>=>>>; where a; b 0 and 0 < a + b 1 ; is c yclic with A 1 = f 2 g and A 2 = f 3 g : Chakraborty and Rao' s result, that in all cases e xcept when S is contained in a group of permutation matrices, the c yclicity property of the elements in S is necessary and suf cient for the sequence n not to con v er ge weakly is the same result obtained by Dhar and Mukherjea e xpressed in more succint terms. One moti v ation for our results in this chapter is to in v estig ate if this connection between c yclicity and weak con v er gence continues to hold e v en for d £ d stochastic matrices, where d > 3 : W e will sho w here that, as long as the common rank of the matrices in the k ernel K is 2 ; the equi v alence of c yclicity of the support S and non-weak con v er gence of n still holds when d > 3 ; b ut the c yclicity property is not necessary for non-weak con v er gence of n if the rank of the matrices in K is bigger than 2 : 9

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2.2 Preliminaries W e be gin with some notations and standard denitions. From no w on, unless otherwise stated, all matrices ha v e real entries and are d £ d; with d 2 : A matrix A is stochastic if its entries are all nonne g ati v e and the sum of the entries in each ro w is 1 : A set S of matrices forms a semigroup if it is closed under matrix multiplication. A semigroup S is said to be left-zero (resp. right-zero ) if AB = A (resp. AB = B ) for all A; B 2 S : A subset X of a semigroup S is called a right ideal if X S X ; where (as usual) X S = f AB : A 2 X ; B 2 S g : A left ideal is dened similarly X is a tw o-sided ideal or simply an ideal if it is simultaneously a leftand right ideal of S : The smallest (relati v e to set inclusion) ideal of S is called its k ernel S is called simple if it has no proper ideals. A matrix A 2 S is idempotent if A 2 = A: If A is idempotent, and, in addition, there is no other idempotent B 2 S satisfying AB = B A = B ; then A is called primiti v e S is called completely simple if it is simple and it contains a primiti v e idempotent. Idempotent stochastic matrices will play a major role in the analysis needed for our problem, so we ne xt present some results that will be used in the sequel. The rst such result is the follo wing well-kno wn structure theorem for idempotent stochastic matrices (see, for e xample, [M2 ]). T H E O R E M 2.1 Let A be a d £ d idempotent stoc hastic matrix. Let p be the r ank of A Then ther e is a partition f T ; C 1 ; C 2 ; : : : ; C p g of f 1 ; 2 ; : : : ; d g ; called a basis of A; suc h that the following hold: 1. j 2 T means that the j th column of A is a zer o column, 2. eac h C k £ C k bloc k of A is a strictly positive bloc k with identical r ows, and 3. eac h C j £ C k ; j 6 = k ; bloc k of A is an all zer o bloc k.Proof.See [M2]. 10

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The set T in a basis f T ; C 1 ; C 2 ; : : : ; C p g of A is called its T -class and the C k s are called its C -classes T w o d £ d idempotent stochastic matrices A and B are called essentially the same if the y ha v e the same rank p and ha v e bases gi v en by: A : f T ; C 1 ; C 2 ; : : : ; C p g ; B : f T 0 ; C 0 1 ; C 0 2 ; : : : ; C 0 p g ; (2. 1) such that for 1 j ; k p; the T £ C k block of A is identical to the T 0 £ C 0 k block of B (for each k ), and the C j £ C k block of A is identical to the C 0 j £ C 0 k block of B for each pair ( j ; k ) : A and B are of the same type if f their bases as gi v en by (2. 1) abo v e are such that j T j = j T 0 j ; j C 1 j = j C 0 1 j ; j C 2 j = j C 0 2 j ; : : : ; j C p j = j C 0 p j ; and furthermore, for each t 2 T ; there is a unique t 0 2 T 0 such that for each k ; 1 k p: the f t g £ C k block of A is a strictly positi v e block if f the f t 0 g £ C 0 k block of B is also so. E X A M P L E 1 The matrices 0BBB@ 0 1 0 0 1 0 0 0 1 1CCCA ; 0BBB@ 0 1 = 2 1 = 2 0 1 0 0 0 1 1CCCA ; and 0BBB@ 0 1 = 3 2 = 3 0 1 0 0 0 1 1CCCA are essentially dif ferent from each other although the second and the third matrices are of the same type. The rst matrix, ho we v er has a type dif ferent from that of the other tw o. The follo wing theorem asserts that as f ar as idempotent matrices are concerned, the concepts of similarity and essential sameness are equi v alent. T H E O R E M 2.2 T wo d £ d stoc hastic idempotent matrices A and B ar e essentially the same if f ther e is a d £ d permutation matrix P suc h that P B = AP :Proof.Suppose A and B are essentially the same. Consider the bases of A and B as gi v en abo v e in (2. 1). Dene the bijection f from f 1 ; 2 ; : : : ; d g onto itself such that f ( T ) = T 0 and f ( C k ) = C 0 k for each k ; and furthermore, for all i; j 2 f 1 ; 2 ; : : : ; d g ; A ij = B f ( i ) ;f ( j ) : 11

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Note that this is clearly possible since A and B are essentially the same. No w dene the permutation matrix P such that P ij = 8<: 1 if j = f ( i ) ; 0 otherwise : Then, for all i; j ; ( P B ) ij = X k P ik B k j = P i;f ( i ) B f ( i ) ;j = B f ( i ) ;j : (2. 2) Write t = f ¡ 1 ( j ) : Then ( AP ) ij = X k A ik P k j = A it P tj = A it ; which is equal to the right hand side of (2. 2) since f ( t ) = j : Thus, P B = AP : Con v ersely e v ery permutation matrix P denes a bijection f as abo v e such that for each i; the element on the i th ro w and f ( i ) th column of P is 1 : This means that the element on the i th ro w and j th column of P is the same as the element on the f ( i ) th ro w and f ( j ) th column of P ¡ 1 AP : Thus, if we dene T 0 = f ( T ) ; and C 0 k = f ( C k ) ; where A has the basis f T ; C 1 ; C 2 ; : : : ; C p g ; then A and B = P ¡ 1 AP are essentially the same. C O R O L L A R Y 2.0.1 T wo d £ d idempotent stoc hastic matrices A and B ar e of the same type if f ther e is a permutation matrix P suc h that the matrices B and P ¡ 1 AP have the same bases.Proof.Immediate from the proof of Theorem 2.2. W e note that in Corollary 2.0.1, B and P ¡ 1 AP may not be essentially the same, despite ha ving the same bases and the same type. E X A M P L E 2 Let us here e xhibit all possible types of idempotent 4 £ 4 stochastic matrices of rank 2 or 3 : 12

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Rank 2 i. Basis T = ; ; C 1 = f 1 g ; C 2 = f 2 ; 3 ; 4 g : 0BBBBBB@ 1 0 0 0 0 a b 1 ¡ a ¡ b 0 a b 1 ¡ a ¡ b 0 a b 1 ¡ a ¡ b 1CCCCCCA where a; b; a + b 2 (0 ; 1) : (2. 3) ii. Basis T = ; ; C 1 = f 1 ; 2 g ; C 2 = f 3 ; 4 g : 0BBBBBB@ a 1 ¡ a 0 0 a 1 ¡ a 0 0 0 0 b 1 ¡ b 0 0 b 1 ¡ b 1CCCCCCA where a; b 2 (0 ; 1) : (2. 4) iii. Basis T = f 1 g ; C 1 = f 2 ; 3 g ; C 2 = f 4 g : 0BBBBBB@ 0 0 0 1 0 a 1 ¡ a 0 0 a 1 ¡ a 0 0 0 0 1 1CCCCCCA where a 2 (0 ; 1) : (2. 5) i v Basis T = f 1 ; 2 g ; C 1 = f 3 g ; C 2 = f 4 g : 0BBBBBB@ 0 0 a 1 ¡ a 0 0 b 1 ¡ b 0 0 1 0 0 0 0 1 1CCCCCCA where a; b 2 [0 ; 1] : (2. 6) Note that (2. 6) gi v es rise to nine dif ferent types according to whether a; b 2 (0 ; 1) ; a = 0 and b 2 (0 ; 1) ; a = 1 and b 2 (0 ; 1) ; b = 0 and a 2 (0 ; 1) ; b = 1 and a 2 (0 ; 1) ; ( a; b ) = (0 ; 0) ; ( a; b ) = (1 ; 0) ; ( a; b ) = (0 ; 1) and ( a; b ) = (1 ; 1) : 13

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Rank 3 i. Basis T = ; ; C 1 = f 1 ; 2 g ; C 2 = f 3 g ; C 3 = f 4 g : 0BBBBBB@ a 1 ¡ a 0 0 a 1 ¡ a 0 0 0 0 1 0 0 0 0 1 1CCCCCCA where a 2 (0 ; 1) ; (2. 7) ii. Basis T = f 1 g ; C 1 = f 2 g ; C 2 = f 3 g ; C 3 = f 4 g and none of the T £ C j ; j = 1 ; 2 ; 3 ; block of a matrix of this type is 0 or 1 : 0BBBBBB@ 0 a b 1 ¡ a ¡ b 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA where a; b; a + b 2 (0 ; 1) ; (2. 8) iii. Same basis as T ype (ii), b ut e xactly one of the T £ C j ; j = 1 ; 2 ; 3 ; block of a matrix of this type is 1; there are three dif ferent types: 0BBBBBB@ 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; or 0BBBBBB@ 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; (2. 9) i v Same basis as T ype (ii), b ut e xactly one of the T £ C j ; j = 1 ; 2 ; 3 ; block of a matrix of this type is 0; here also there are three dif ferent types: 0BBBBBB@ 0 0 a 1 ¡ a 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ 0 a 0 1 ¡ a 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; or 0BBBBBB@ 0 a 1 ¡ a 0 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; (2. 10) where a 2 (0 ; 1) : 14

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E X A M P L E 3 Let us no w consider the general form of an idempotent d £ d stochastic matrix e of rank 2 : Let f T ; C 1 ; C 2 g be a basis of e: Let k = j T j ; c 1 = j C 1 j ; and c 2 = j C 2 j ; where k + c 1 + c 2 = d: According to Theorem 2.1, e must ha v e the block form e = 0BBB@ 0 A 0 B 0 0 A 0 0 0 B 1CCCA ; (2. 11) where the top left zero block is k £ k ; and A (resp. B ) is a c 1 £ c 1 (resp. c 2 £ c 2 ) strictly positi v e stochastic matrix with identical ro ws, each equal to A 1 = ( a 1 ; a 2 ; : : : ; a c 1 ) (resp. B 1 = ( b 1 ; b 2 ; : : : ; b c 2 ) ). Since e has rank 2 ; each of the rst k ro ws of e must be a linear combination of A 1 and B 1 : In other w ords, the k £ c 1 matrix A 0 and the k £ c 2 matrix B 0 must ha v e the form A 0 = 0BBBBBB@ r 1 A 1 r 2 A 1 ... r k A 1 1CCCCCCA (2. 12) and B 0 = 0BBBBBB@ (1 ¡ r 1 ) B 1 (1 ¡ r 2 ) B 1 ... (1 ¡ r k ) B 1 1CCCCCCA ; (2. 13) for some constants r 1 ; r 2 ; : : : ; r k 2 [0 ; 1] : In the sequel, we will also need the follo wing result concerning nite groups of d £ d stochastic matrices of common rank. T H E O R E M 2.3 Let G be a nite gr oup of d £ d stoc hastic matrices of common r ank p: Let f T ; C 1 ; C 2 ; : : : ; C p g be the basis of the identity of G: Then ther e an isomorphism fr om G to a subgr oup of the gr oup S p of permutations on f 1 ; 2 ; : : : ; p g suc h that if is the isomorphic ima g e of A 2 G; then the C j £ C k bloc k of A is an all-zer o bloc k whene ver ( j ) 6 = k : 15

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Proof.See [M2]. E X A M P L E 4 In Theorem 2.3, consider the case when p = 2 in the general d £ d situation. The isomorphism from G to S 2 has e gi v en by (2. 11) in Example 3 as the isomorphic preimage of the identity permutation. Here, we calculate the d £ d stochastic matrix preimage y of the permutation (12) under this isomorphism. Let us follo w the denitions and notations in Example 3. Then, immediately from Theorem 2.3, we kno w that the C 1 £ C 1 and C 2 £ C 2 blocks of y must be zero blocks. Further from the equation y e = y ; it follo ws that the rst k columns of y must be zero columns. Write y = 0BBB@ 0 U V 0 0 W 0 Z 0 1CCCA ; where U is k £ c 1 ; V is k £ c 2 ; W is c 1 £ c 2 ; and Z is c 2 £ c 1 ; with W and Z both stochastic. The equations y e = y ; ey = y ; and y 2 = e then translate to the matrix equations U A = U ; V B = V ; W B = W ; Z A = Z ; (2. 14) B 0 Z = U ; A 0 W = V ; AW = W ; B Z = Z ; (2. 15) and V Z = A 0 ; U W = B 0 ; W Z = A; Z W = B ; (2. 16) respecti v ely Since both A and B ha v e identical ro ws, and therefore constant columns, the last tw o equations in (2. 15) imply that W and Z do, too. This information, together with the last tw o equations in (2. 14), completely determine W and Z : W = 0BBBBBB@ B 1 B 1 ... B 1 1CCCCCCA ( c 1 -man y ro ws ) and Z = 0BBBBBB@ A 1 A 1 ... A 1 1CCCCCCA ( c 2 -man y ro ws ) : 16

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The rst equation in (2. 15) then implies that the entry in the j th ro w and l th column of U ; where j 2 T and l 2 C 1 ; is X s 2 C 2 (1 ¡ r j ) b s a l = (1 ¡ r j ) a l ; while the second equation in (2. 15) implies that the entry in the j th ro w and l th column of V ; where j 2 T and l 2 C 2 ; is X s 2 C 1 r j a s b l = r j b l : Thus, y has the block form y = 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 0 (1 ¡ r 1 ) A 1 (1 ¡ r 2 ) A 1 ... (1 ¡ r k ) A 1 r 1 B 1 r 2 B 1 ... r k B 1 0 0 B 1 B 1 ... B 1 0 A 1 A 1 ... A 1 0 1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (2. 17) Ne xt we present the follo wing general theorem which forms the basis for the method presented in [DM]: T H E O R E M 2.4 Let S be a locally compact second countable Hausdorf f semigr oup and be a pr obability measur e on the Bor el subsets of S : Suppose that S = 1 [ n =1 S n ; wher e S is the support of : 17

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Suppose that the sequence f n : n 1 g is a tight sequence; that is, given > 0 ; ther e is a compact set K suc h that for all n 1 ; n ( K ) > 1 ¡ ": Then the sequence (1 =n ) P nk =1 k con ver g es weakly to a pr obability measur e ; wher e S ; the support of ; is the k ernel K of S : The gr oup factor G of K (whic h is completely simple) is compact. The sequence n con ver g es weakly to if f ther e does not e xist a subgr oup H of K suc h that the following conditions hold: 1. H is a normal subgr oup of the gr oup eK e G; wher e e is the identity of H ; 2. Y X H ; wher e Y is the set of all idempotents in K e and X is the set of all idempotent elements in eK ; 3. eS e g H for some g 2 G n H :Proof.See Theorem 2.1 in [LM]. W e note that when S is contained in a set of d £ d stochastic matrices, which is the case of interest to us, the most important assertion in Theorem 2.4 rele v ant to our problem is the follo wing: the sequence n does not con v er ge weakly if and only if there e xists an idempotent e in the k ernel K of S such that eS e g H ; (2. 18) for some g 2 eK e n H and some proper normal subgroup H of eK e: W e remark that Dhar and Mukherjea' s solution in [DM ] is essentially accomplished by translating (2. 18) into a set of conditions on S by considering the possibilities for the k ernel K and the proper normal subgroups of the corresponding compact group eK e: F or emphasis, we record the follo wing information which follo ws when Theorem 2.4 is applied to the case under in v estig ation. The k ernel K ; structure wise, is a completely simple semigroup; in other w ords, K is topologically isomorphic to (that is, can be identied with) the product X £ G £ Y ; where G is a nite group and G = eK e; where e is some x ed idempotent matrix in K ; X is a left-zero semigroup consisting of all idempotent matrices in 18

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K e; Y is a right-zero semigroup consisting of all idempotent matrices in eK ; and Y X G: The multiplication in X £ G £ Y is gi v en by: ( x; g ; y )( x 0 ; g 0 ; y 0 ) = ( x; g ( y x 0 ) g 0 ; y 0 ) : Details of these are also gi v en in [HMu ]. As mentioned in Section 2.1, it is well-kno wn that K consists of all matrices in S which ha v e the minimal rank. Moreo v er if this minimal rank is one (that is, the matrices in the k ernel K ha v e identical ro ws), then Lemma 2.1 belo w says that the sequence n al w ays con v er ges weakly L E M M A 2.1 Let be a pr obability measur e with support S inside a set of d £ d stoc hastic matrices. Let K be the k ernel of the closed semigr oup S g ener ated by S : If the common r ank of the matrices in K is equal to 1, then the sequence n con ver g es weakly .Proof.It is easy to v erify that for an y tw o stochastic matrices A and B in K ; AB = B : Notice that since S is compact, the a v eraged sequence (1 =n ) P nk =1 k al w ays con v er ges to some probability measure ; whose support S is actually equal to K : Further for an y open set G containing K ; n ( G ) 1 as n 1 : This means that if 0 is another weak limit point of ( n ) ; then its support S 0 is inside K : Since ¤ = ; it follo ws that ¤ ( n ) = for each n; so ¤ 0 = : But ¤ 0 = 0 since AB = B for an y tw o matrices A; B 2 K : Thus, 0 = ; and so n con v er ges weakly to : Also, when the rank of the matrices in K is d (that is, when the y all ha v e full rank), then K happens to be a compact group, and in this case, K = S consists of d £ d in v ertible matrices, and as such, the question of weak con v er gence of n can be easily resolv ed using well-kno wn classical results. Thus, we will only need to look into the cases when K consists of matrices with rank r ; where r is strictly between 1 and d: F or the special case when Y X = G in Theorem 2.4, we ha v e the follo wing: C O R O L L A R Y 2.1.1 Let be a pr obability measur e on the Bor el subsets of d £ d stoc hastic matrices, and let S = ¡ S n 1 S n ¢ be the closed (multiplicative) semigr oup g ener ated by the support S of : Let K be the k ernel of S and let X £ G £ Y be the pr oduct r epr esentation of K : If Y X = G; then n con ver g es weakly 19

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Proof.Let be the identity of the group of weak limit points of n If Y X = G; then by Theorem 2.4, the support S of has the product representation X £ G £ Y ; the same as that of S ; where = ( w ) lim n !1 (1 =n ) P nk =1 k : W e also ha v e ¤ = ¤ = ; (2. 19) since ¤ = ¤ = : By (2. 19) and Proposition 2.5 (page 74, [HMu ]) it follo ws that for an y Borel set B K and an y x 2 K ; f y : y x 2 B g = Z f z : z y x 2 B g ( dy ) = f y : y x 2 B g ; and since = ¤ and = ¤ ; we also ha v e ( B ) = Z f y : y x 2 B g ( dx ) = Z f y : y x 2 B g ( dx ) = ¤ ( B ) = ( B ) : Thus, = : It follo ws that ¤ = ¤ = : (2. 20) This means that for an y weak limit point 0 of n ; 0 = 0 ¤ = ¤ 0 = : In other w ords, whene v er Y X = G; n con v er ges weakly to : Ne xt we dene c yclicity follo wing [CR ] (see page 169, at the end of the paper). Let A 1 ; A 2 ; : : : ; A k be pairwise disjoint subsets of f 1 ; 2 ; : : : ; d g so that [ ki =1 A i may or may not equal f 1 ; 2 ; : : : ; d g : Then S is called c yclic with respect to f A 1 ; A 2 ; : : : ; A k g if, for each x in S with A k + i A i ; 1 i k ; we ha v e X j 2 A m +1 x ij = 1 ; i 2 A m ; 1 m k : (2. 21) This denition immediately gi v es us the follo wing: 20

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L E M M A 2.2 Let be a pr obability measur e on the Bor el subsets of d £ d stoc hastic matrices, and let S be the support of : If S is cyclic, then n does not con ver g e weakly .Proof.Dene the sets C and D as C = f x 2 S : X j 2 A 1 x ij = 1 for each i 2 A 2 g and D = f x 2 S : X j 2 A 2 x ij = 1 for each i 2 A 1 g ; where A 1 and A 2 are as the y appear in the denition of c yclicity abo v e, then C and D are disjoint compact subsets of S ; and furthermore, for each n 1 ; nk ( C ) = 1 and nk +1 ( D ) = 1 : If n con v er ges weakly to as n 1 ; then clearly ( C ) = 1 as well as ( D ) = 1 : But this is impossible. Thus, c yclicity of S implies non-weak con v er gence of n : In the ne xt section, we e xplicitly solv e the problem of weak con v er gence in 4 £ 4 stochastic matrices follo wing Dhar and Mukherjea' s method in [DM]. As pointed out at the end of the introduction to the present chapter one reason for this calculation is to v erify whether Chakraborty and Rao' s characterization, in the 3 £ 3 case, of non-weak con v er gence of the sequence n in terms of c yclicity of the support S is still v alid in the 4 £ 4 case. 2.3 4 £ 4 Stochastic Matrices In this section, is a probability measure with support S inside a set of 4 £ 4 stochastic matrices, S is the multiplicati v e semigroup generated by S ; and K is the k ernel of S : Our aim here is to use (2. 18) in Section 2.2 to nd a necessary and suf cient condition on S in order for the sequence n not to con v er ge weakly Suppose, then, that in e v erything that follo ws, n does not con v er ge weakly First, let the common rank of the matrices in K be 2 : According to Theorem 2.3, the compact group eK e of Theorem 2.4 must then be isomorphic to the tw o-element symmetric 21

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group S 2 = f (1) ; (12) g : Here, e is one of the idempotent 4 £ 4 stochastic matrices of rank 2 gi v en by (2. 3), (2. 4), (2. 5), or (2. 6) in Example 2. Suppose e has the form displayed in (2. 3). That is, e = 0BBBBBB@ 1 0 0 0 0 a b 1 ¡ a ¡ b 0 a b 1 ¡ a ¡ b 0 a b 1 ¡ a ¡ b 1CCCCCCA where a; b; a + b 2 (0 ; 1) : Since the C -classes of e are C 1 = f 1 g and C 2 = f 2 ; 3 ; 4 g ; the C 1 £ C 1 and C 2 £ C 2 blocks of the stochastic matrix A 2 eK e corresponding to the permutation = (12) are zero blocks, according to Theorem 2.3. Therefore, A must ha v e the form 0BBBBBB@ 0 s t 1 ¡ s ¡ t 1 0 0 0 1 0 0 0 1 0 0 0 1CCCCCCA for some s; t 2 [0 ; 1] : The equation A 2 = e then sho ws that s = a and t = b; gi ving A = 0BBBBBB@ 0 a b 1 ¡ a ¡ b 1 0 0 0 1 0 0 0 1 0 0 0 1CCCCCCA : (2. 22) The only proper normal subgroup of eK e = f e; A g is the tri vial subgroup f e g ; and consequently its coset is the singleton f A g : Thus, the condition (2. 18) translates to the equation exe = A; (2. 23) where x = 0BBBBBB@ a 1 a 2 a 3 1 ¡ a 1 ¡ a 2 ¡ a 3 b 1 b 2 b 3 1 ¡ b 1 ¡ b 2 ¡ b 3 c 1 c 2 c 3 1 ¡ c 1 ¡ c 2 ¡ c 3 d 1 d 2 d 3 1 ¡ d 1 ¡ d 2 ¡ d 3 1CCCCCCA (2. 24) 22

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is an arbitrary element of the support S : W e note that in (2. 24), all v ariables are nonne gati v e and at most equal to 1 : (2. 23) then leads to the equations a 1 = 0 and ab 1 + bc 1 + (1 ¡ a ¡ b ) d 1 = 1 : (2. 25) It follo ws from (2. 25) that b 1 = c 1 = d 1 = 1 ; leading to x = 0BBBBBB@ 0 s t 1 ¡ s ¡ t 1 0 0 0 1 0 0 0 1 0 0 0 1CCCCCCA : W e conclude that under the condition of non-weak con v er gence of the sequence n ; and with e gi v en by (2. 3), the support S must satisfy S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s t 1 ¡ s ¡ t 1 0 0 0 1 0 0 0 1 0 0 0 1CCCCCCA : s; t; s + t 2 [0 ; 1] 9>>>>>>=>>>>>>; : (2. 26) Recalling Chakraborty and Rao' s denition of c yclicity we conclude that S in (2. 26) is c yclic with respect to the disjoint subsets f 1 g and f 2 ; 3 ; 4 g of f 1 ; 2 ; 3 ; 4 g : In the follo wing, when describing a condition of set inclusion that S must satisfy for non-weak con v er gence of the sequence n ; as in (2. 26), if doing so will not cause confusion we will omit the description of the v ariables used, with the understanding that the y are all non-ne g ati v e and at most 1 ; and such that the matrix the y form is stochastic. Similar calculations sho w that if e is gi v en by (2. 4), that is, e = 0BBBBBB@ a 1 ¡ a 0 0 a 1 ¡ a 0 0 0 0 b 1 ¡ b 0 0 b 1 ¡ b 1CCCCCCA ; 23

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where a; b; 2 (0 ; 1) ; then S must satisfy S 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 s 1 ¡ s 0 0 t 1 ¡ t u 1 ¡ u 0 0 v 1 ¡ v 0 0 1CCCCCCA : s; t; u; v 2 [0 ; 1] 9>>>>>>=>>>>>>; ; (2. 27) in which case S is c yclic with respect to f 1 ; 2 g and f 3 ; 4 g : Let us ne xt consider the case when e is gi v en by (2. 5), that is, e = 0BBBBBB@ 0 0 0 1 0 a 1 ¡ a 0 0 a 1 ¡ a 0 0 0 0 1 1CCCCCCA ; where a 2 (0 ; 1) : In this case, the C -classes are C 1 = f 2 ; 3 g and C 2 = f 4 g ; so that if eK e = f e; A g ; then A = 0BBBBBB@ 0 s 1 ¡ s 0 0 0 0 1 0 0 0 1 0 a 1 ¡ a 0 1CCCCCCA ; (2. 28) where s 2 [0 ; 1] : It then follo ws that elements in S must be of the form 0BBBBBB@ s 1 s 2 s 3 1 ¡ s 1 ¡ s 2 ¡ s 3 t 0 0 1 ¡ t u 0 0 1 ¡ u 0 v 1 ¡ v 0 1CCCCCCA ; (2. 29) which may be simplied further as follo ws. Pick an element y of the form gi v en by (2. 29). Then with e as abo v e, ey 2 e = 0BBBBBB@ 0 0 0 1 0 ¤ ¤ (1 ¡ ( s 2 + s 3 ))( at + (1 ¡ a ) u ) 0 ¤ ¤ (1 ¡ ( s 2 + s 3 ))( at + (1 ¡ a ) u ) 0 0 0 1 1CCCCCCA : 24

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By inspection, this element cannot be A in (2. 28), so it must be e itself, implying that (1 ¡ ( s 2 + s 3 )) ( at + (1 ¡ a ) u ) = 0 : Thus, if there e xists y 2 S with at least one of t; u > 0 ; then s 2 + s 3 = 1 (since a 2 (0 ; 1) ) and accordingly S must satisfy S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 1 ¡ s 0 t 0 0 1 ¡ t u 0 0 1 ¡ u 0 v 1 ¡ v 0 1CCCCCCA : s; v 2 [0 ; 1] and either t or u or both are positi v e 9>>>>>>=>>>>>>; ; (2. 30) otherwise t = u = 0 and S must satisfy S 8>>>>>><>>>>>>: 0BBBBBB@ s 1 s 2 s 3 1 ¡ s 1 ¡ s 2 ¡ s 3 0 0 0 1 0 0 0 1 0 v 1 ¡ v 0 1CCCCCCA 9>>>>>>=>>>>>>; : (2. 31) Note that in (2. 30), S is c yclic with respect to f 1 ; 4 g and f 2 ; 3 g ; and in (2. 31), S is c yclic with respect to f 2 ; 3 g and f 4 g : T o nish the rank 2 case, we consider the situation when e is gi v en by (2. 6), that is, e = 0BBBBBB@ 0 0 a 1 ¡ a 0 0 b 1 ¡ b 0 0 1 0 0 0 0 1 1CCCCCCA ; where a; b 2 [0 ; 1] : Here, the appropriate condition on S depends on a and b: W e gi v e all possibilities belo w and the corresponding condition on S : a; b 2 (0 ; 1) : S 8>>>>>><>>>>>>: 0BBBBBB@ s 1 s 2 s 3 1 ¡ s 1 ¡ s 2 ¡ s 3 t 1 t 2 t 3 1 ¡ t 1 ¡ t 2 ¡ t 3 0 0 0 1 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; ; (2. 32) 25

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and S is c yclic with respect to f 3 g ; f 4 g : a = 0 and b 2 (0 ; 1) : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 1 0 t 1 t 2 t 3 1 ¡ t 1 ¡ t 2 ¡ t 3 u 0 0 1 ¡ u 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 33) for some x ed u > 0 ; or else S satises the same condition as (2. 32). Note that S in (2. 33) is c yclic with respect to f 1 ; 4 g ; f 3 g : a = 1 and b 2 (0 ; 1) : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 0 1 t 1 t 2 t 3 1 ¡ t 1 ¡ t 2 ¡ t 3 0 0 0 1 u 0 1 ¡ u 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 34) for some x ed u > 0 ; or else S satises the same condition as (2. 32). Note that S in (2. 34) is c yclic with respect to f 1 ; 3 g ; f 4 g : a 2 (0 ; 1) and b = 0 : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ s 1 s 2 s 3 1 ¡ s 1 ¡ s 2 ¡ s 3 0 0 1 0 0 u 0 1 ¡ u 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 35) for some x ed u > 0 ; or else S satises the same condition as (2. 32). Note that S in (2. 35) is c yclic with respect to f 2 ; 4 g ; f 3 g : 26

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a 2 (0 ; 1) and b = 1 : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ s 1 s 2 s 3 1 ¡ s 1 ¡ s 2 ¡ s 3 0 0 0 1 0 0 0 1 0 u 1 ¡ u 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 36) for some x ed u > 0 ; or else S satises the same condition as (2. 32). Note that S in (2. 36) is c yclic with respect to f 2 ; 3 g ; f 4 g : a = b = 0 : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 1 0 0 0 1 0 u v 0 1 ¡ u ¡ v 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 37) for some x ed u; v > 0 ; or else S satises the same condition as (2. 32), (2. 33), or (2. 35). Note that S in (2. 37) is c yclic with respect to f 1 ; 2 ; 4 g ; f 3 g : a = 1 ; b = 0 : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 0 1 ¡ s t 0 1 ¡ t 0 0 u 0 1 ¡ u 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 38) for x ed u; t > 0 ; or S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 0 1 ¡ s t 0 1 ¡ t 0 0 0 0 1 v 0 1 ¡ v 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 39) 27

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for x ed s; v > 0 ; or S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 0 1 ¡ s t 0 1 ¡ t 0 0 u 0 1 ¡ u v 0 1 ¡ v 0 1CCCCCCA 9>>>>>>=>>>>>>; : (2. 40) for x ed u; v > 0 ; or else S satises the same condition as (2. 32), (2. 34), or (2. 35). Note that S in (2. 38), (2. 39), or (2. 40) is c yclic with respect to f 1 ; 3 g ; f 2 ; 4 g : a = 0 ; b = 1 : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 1 ¡ s 0 t 0 0 1 ¡ t u 0 0 1 ¡ u 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 41) for x ed s; u > 0 ; or S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 1 ¡ s 0 t 0 0 1 ¡ t 0 0 0 1 0 v 1 ¡ v 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 42) for x ed t; v > 0 ; or S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 s 1 ¡ s 0 t 0 0 1 ¡ t u 0 0 1 ¡ u 0 v 1 ¡ v 0 1CCCCCCA 9>>>>>>=>>>>>>; : (2. 43) for x ed u; v > 0 ; or else S satises the same condition as (2. 32), (2. 33), or (2. 36). Note that S in (2. 41), (2. 42), or (2. 43) is c yclic with respect to f 1 ; 4 g ; f 2 ; 3 g : 28

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a = b = 1 : Either S satises S 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 0 1 0 0 0 1 0 0 0 1 u v 1 ¡ u ¡ v 0 1CCCCCCA 9>>>>>>=>>>>>>; ; (2. 44) for some x ed u; v > 0 ; or else S satises the same condition as (2. 32), (2. 34), or (2. 36). Note that S in (2. 44) is c yclic with respect to f 1 ; 2 ; 3 g ; f 4 g : This completes the rank 2 case. W e point out that in all cases, n not weakly con v er gent implies S is c yclic. W e no w assume that the common rank of the matrices in the k ernel K is 3 : In this case, the group eK e; where e is a 4 £ 4 idempotent stochastic matrix of rank 3 ; is isomorphic to a subgroup of the symmetric group S 3 of permutations on f 1 ; 2 ; 3 g : Note that the nontri vial proper subgroups of S 3 are the three tw o-element subgroups each consisting of the identity permutation and a transposition, together with the normal subgroup consisting of the e v en permutations. It is therefore clear that the condition which S must satisfy under the same assumption that the sequence n does not con v er ge weakly is eS e f M (12) ; M (13) ; M (23) g (2. 45) (assuming eS e has more than one element), where M is the stochastic matrix pre-image of the transposition 2 f (12) ; (13) ; (23) g ; under the isomorphism of Theorem 2.3. As w as done in the rank 2 case, we let x = 0BBBBBB@ a 1 a 2 a 3 1 ¡ a 1 ¡ a 2 ¡ a 3 b 1 b 2 b 3 1 ¡ b 1 ¡ b 2 ¡ b 3 c 1 c 2 c 3 1 ¡ c 1 ¡ c 2 ¡ c 3 d 1 d 2 d 3 1 ¡ d 1 ¡ d 2 ¡ d 3 1CCCCCCA be an arbitrary element of the support S ; and solv e the equation exe = M (2. 46) 29

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arising from (2. 45), for each of the possibilities for e gi v en by (2. 7), (2. 8), (2. 9), or (2. 10), and the corresponding stochastic matrices M ; = (12) ; (13) ; (23) : W e sho w the details of the calculations in v olv ed using one particular type of a 4 £ 4 idempotent stochastic matrix e of rank 3 ; namely the rst matrix displayed in (2. 9). That is, e = 0BBBBBB@ 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA : From Theorem 2.3, the stochastic matrices M (12) ; M (13) ; and M (23) in this case are gi v en by 0BBBBBB@ 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1CCCCCCA ; and 0BBBBBB@ 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1CCCCCCA ; respecti v ely Then it follo ws from the equality exe = M (12) (after computations) that x must be of the form 0BBBBBB@ s t u v 0 0 1 0 w 1 ¡ w 0 0 0 0 0 1 1CCCCCCA where e v ery entry is nonne g ati v e with each ro w sum equal to 1 : Let us call a typical element of this form x 1 : Noting that the element ex 21 e is an element in eK e; and after computations looking at its form, it is clear that this element must be the element e; whence it follo ws 30

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(after computations) that x 1 must be one of the follo wing tw o possible forms: 0BBBBBB@ s 0 t 0 u 0 v 0 0 0 1 0 0 1 0 0 0 0 0 1 1CCCCCCA or 0BBBBBB@ 0 0 1 0 0 0 1 0 c 1 ¡ c 0 0 0 0 0 1 1CCCCCCA ; where ag ain the entries are nonne g ati v e with each ro w sum 1 : Let us call a typical element of the rst form y and a typical element of the second form z : Noting that if there e xists a z with c > 0 ; then since e ( z y ) e must ag ain equal e; it follo ws after computations that in y ; c 0 = 1 if there is a z in S with c > 0 : In other w ords, in the case when exe = M (12) ; the form of x must be that of z abo v e where c 0 : By also considering the equations exe = M (13) and exe = M (23) ; and going through similar ar guments and computations, we see that S must be contained in the set 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 1 0 0 0 1 0 c 1 ¡ c 0 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ 0 0 0 1 0 0 0 1 0 0 1 0 d 1 ¡ d 0 0 1CCCCCCA ; 0BBBBBB@ a 1 ¡ a 0 0 b 1 ¡ b 0 0 0 0 0 1 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; : (2. 47) Note that S in this case cannot be c yclic if it has more than one element. The corresponding conditions for S when e is gi v en by the other tw o matrices displayed in (2. 9) are S 8>>>>>><>>>>>>: 0BBBBBB@ 0 1 0 0 f 0 1 ¡ f 0 0 1 0 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ u 0 1 ¡ u 0 0 0 0 1 g 0 1 ¡ g 0 0 1 0 0 1CCCCCCA ; 0BBBBBB@ 0 0 0 1 0 1 0 0 0 0 0 1 h 0 1 ¡ h 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 48) and S 8>>>>>><>>>>>>: 0BBBBBB@ r 0 0 1 ¡ r 0 0 1 0 0 1 0 0 f 0 0 1 ¡ f 1CCCCCCA ; 0BBBBBB@ 0 1 0 0 g 0 0 1 ¡ g 0 0 1 0 0 1 0 0 1CCCCCCA ; 0BBBBBB@ 0 0 1 0 0 1 0 0 h 0 0 1 ¡ h 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; ; (2. 49) 31

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respecti v ely Ne xt, when e has the form gi v en by (2. 7), that is, e = 0BBBBBB@ a 1 ¡ a 0 0 a 1 ¡ a 0 0 0 0 1 0 0 0 0 1 1CCCCCCA : where a 2 (0 ; 1) ; then S must satisfy S 8>>>>>><>>>>>>: 0BBBBBB@ 0 0 1 0 0 0 1 0 u 1 ¡ u 0 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ 0 0 0 1 0 0 0 1 0 0 1 0 v 1 ¡ v 0 0 1CCCCCCA ; 0BBBBBB@ s 1 ¡ s 0 0 t 1 ¡ t 0 0 0 0 0 1 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; (2. 50) and when e is gi v en by (2. 8), that is, e = 0BBBBBB@ 0 a b 1 ¡ a ¡ b 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA where a; b; a + b 2 (0 ; 1) ; the condition for S is S 8>>>>>><>>>>>>: 0BBBBBB@ r s t 1 ¡ r ¡ s ¡ t 0 0 1 0 0 1 0 0 0 0 0 1 1CCCCCCA ; 0BBBBBB@ u v w 1 ¡ u ¡ v ¡ w 0 0 0 1 0 0 1 0 0 1 0 0 1CCCCCCA ; 0BBBBBB@ x y z 1 ¡ x ¡ y ¡ z 0 1 0 0 0 0 0 1 0 0 1 0 1CCCCCCA 9>>>>>>=>>>>>>; : (2. 51) Finally when e tak es each of the possible v alues in (2. 10), the same condition gi v en in (2. 51) is obtained. 32

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This nishes our analysis of the rank 2 and rank 3 cases. In vie w of Lemma 2.1 in Section 2.2 and the remarks follo wing its proof, these are the only cases that need to be considered in studying problem of weak con v er gence in 4 £ 4 stochastic matrices. W e therefore ha v e the follo wing: T H E O R E M 2.5 Let be a pr obability measur e on the Bor el subsets of 4 £ 4 stoc hastic matrices suc h that the minimal r ank of the matrices in the k ernel K of the closed semigr oup g ener ated by the support S of is 2 : Then the sequence of con volution power s n does not con ver g e weakly if f S is cyclic. If the common r ank of the matrices in K is 3 ; then cyclicity of S is not necessary for non-weak con ver g ence of n :Proof.F ollo ws from the preceding discussion. W e no w present a couple of e xamples. E X A M P L E 5 Consider a probability measure with support S gi v en by S = f A; B g ; where A = 0BBBBBB@ 0 0 1 0 1 2 0 0 1 2 1 2 0 0 1 2 0 1 0 0 1CCCCCCA and B = 0BBBBBB@ 0 1 0 0 1 2 0 0 1 2 1 2 0 0 1 2 0 0 1 0 1CCCCCCA : (2. 52) Then the semigroup S generated by S has 4 elements and is gi v en by S = f e; A; B ; C g where A and B are gi v en in (2. 52), e = 0BBBBBB@ 1 2 0 0 1 2 0 1 2 1 2 0 0 1 2 1 2 0 1 2 0 0 1 2 1CCCCCCA and C = 0BBBBBB@ 0 1 2 1 2 0 1 2 0 0 1 2 1 2 0 0 1 2 0 1 2 1 2 0 1CCCCCCA : 33

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The multiplication table of S is gi v en by e A B C e e C C C A C e e e B C e e e C C e e e It is clear from the multiplication table that the k ernel K is f e; C g ; which consists precisely of the matrices of the minimal rank 2 : Moreo v er K = eK e is a tw o-element group, and, as such, only has the tri vial subgroup f e g as its normal subgroup. The coset f C g is e xactly the set eS e; and therefore the sequence n does not con v er ge weakly Notice that S is c yclic with respect to the partition f 1 ; 4 g and f 2 ; 3 g of f 1 ; 2 ; 3 ; 4 g : E X A M P L E 6 Consider a tw o-point probability measure with support S gi v en by S = f A; B g ; where A = 0BBBBBB@ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1CCCCCCA and B = 0BBBBBB@ 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1CCCCCCA : (2. 53) Then it can be v eried that the semigroup S generated by S has 8 elements and is gi v en by S = f e; A; B ; C ; D ; E ; F ; I g ; where A and B are gi v en in (2. 53), e = 0BBBBBB@ 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA ; 34

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C = 0BBBBBB@ 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1CCCCCCA ; D = 0BBBBBB@ 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1CCCCCCA ; E = 0BBBBBB@ 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1CCCCCCA ; F = 0BBBBBB@ 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1CCCCCCA ; and I is the 4 £ 4 identity matrix. The multiplication table of S is gi v en by e A B C D E F I e e E B C D E F e A E I D F B e C A B B C e E F C D B C C B F D e B E C D D F E e C F B D F E e D F B e C E F F D C B E D e F I e A B C D E F I Except for A and I ; both of which are of rank 4 ; all the matrices in S ha v e rank 3 : The k ernel K is therefore K = f e; B ; C ; D ; E ; F g : Notice that K = eK e is a group isomorphic to the symmetric group S 3 on f 1 ; 2 ; 3 g ; and the coset f B ; E ; F g of the normal subgroup f e; C ; D g contains eS e = f B ; E g : Thus, the sequence n does not con v er ge weakly Notice also that S is not c yclic. 35

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2.4 The d £ d case Let us no w attempt to solv e the present problem in the general d £ d case. Thus, we let be a probability measure with support S inside a set of d £ d stochastic matrices, S the multiplicati v e semigroup generated by S ; and K the k ernel of S : The rst step is to determine what happens when the rank of the matrices in K is 2 : In this case, the group G = eK e; where e is a x ed idempotent in K ; and X £ G £ Y is a product representation of K (as described in the remarks after Theorem 2.4 of Section 2.2), has one or tw o elements. In case G has only one element, then since Y X G; Y X must indeed equal G; and by Corollary 2.1.1, the sequence n must then con v er ge weakly Thus, if we assume that n does not con v er ge weakly then the group G = eK e has tw o elements f e; y g ; where e is a d £ d idempotent stochastic matrix that has the general form calculated in Example 3 of Section 2.2, and y is the d £ d stochastic matrix satisfying ey = y e = y and y 2 = e: That is, e has basis f T ; C 1 ; C 2 g ; where T ; C 1 ; and C 2 ha v e, respecti v ely k ; c 1 ; and c 2 elements with k + c 1 + c 2 = d; and e = 0BBB@ 0 A 0 B 0 0 A 0 0 0 B 1CCCA : In this block form, the top left zero block is k £ k ; the strictly positi v e rank one stochastic matrices A and B are c 1 £ c 1 and c 2 £ c 2 ; respecti v ely A 0 is k £ c 1 and B 0 is k £ c 2 : If A has identical ro ws equal to A 1 = ( a 1 ; a 2 ; : : : ; a c 1 ) and B has identical ro ws equal to B 1 = ( b 1 ; b 2 ; : : : ; b c 2 ) ; 36

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then A 0 and B 0 are gi v en by A 0 = 0BBBBBB@ r 1 A 1 r 2 A 1 ... r k A 1 1CCCCCCA and B 0 = 0BBBBBB@ (1 ¡ r 1 ) B 1 (1 ¡ r 2 ) B 1 ... (1 ¡ r k ) B 1 1CCCCCCA ; for some constants r 1 ; r 2 ; : : : ; r k 2 [0 ; 1] : W e also kno w from Example 4 of Section 2.2 that y must then ha v e the form y = 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 0 (1 ¡ r 1 ) A 1 (1 ¡ r 2 ) A 1 ... (1 ¡ r k ) A 1 r 1 B 1 r 2 B 1 ... r k B 1 0 0 B 1 B 1 ... B 1 0 A 1 A 1 ... A 1 0 1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; where the diagonal blocks are zero blocks of size k £ k ; c 1 £ c 1 ; and c 2 £ c 2 ; respecti v ely W e no w proceed to determine the block structure of an arbitrary stochastic matrix x = 0BBB@ X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 1CCCA (2. 54) in S : Here, the diagonal blocks X j j ; j = 1 ; 2 ; 3 ; are of size k £ k ; c 1 £ c 1 ; and c 2 £ c 2 ; respecti v ely (The blocks corresponding to x 0 in S will accordingly be referred to as X 0 ij ; those for x 00 will be referred to as X 00 ij ; etc.) 37

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Since we are assuming non-weak con v er gence of n ; exe must be equal to the nonidentity element in the group eK e: In other w ords, eS e = f y g ; (2. 55) which implies, in particular that A ( X 21 A 0 + X 22 A ) = 0 (2. 56) and B ( X 31 B 0 + X 33 B ) = 0 : (2. 57) Since both A and B are strictly positi v e, it follo ws from (2. 56) and (2. 57) that X 22 = 0 (2. 58) and X 33 = 0 ; (2. 59) which consequently gi v e AX 21 A 0 = 0 (2. 60) and B X 31 B 0 = 0 : (2. 61) No w dene the sets T 1 = f t 2 T : r t = 0 g T 2 = f t 2 T : r t = 1 g T 3 = f t 2 T : r t 2 (0 ; 1) g : 9>>>=>>>; (2. 62) F or s; t 2 C 1 ; the ( s; t ) -entry of AX 21 A 0 in (2. 60) is 0 : This entry is ( AX 21 A 0 ) st = X u 2 C 1 X w 2 T a u ( X 21 ) uw r w a t = a t X u 2 C 1 X w 2 T 2 [ T 3 a u ( X 21 ) uw r w ; 38

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implying that ( X 21 ) uw = 0 for all u 2 C 1 and w 2 T 2 [ T 3 : Thus, X 21 j C 1 £ ( T 2 [ T 3 ) = 0 : (2. 63) Similarly from (2. 61) we obtain X 31 j C 2 £ ( T 1 [ T 3 ) = 0 : (2. 64) Ne xt, we kno w that ex 2 e must be either e or y : In the 4 £ 4 rank 2 case, as we sa w in the preceding section, this is easy to determine by looking at the form of ex 2 e and comparing with e or y ; nding that in all cases, ex 2 e = e: In the general case, it is still true that ex 2 e = e; and, in f act, we ha v e follo wing: L E M M A 2.3 Let be a pr obability measur e with support S inside a set of d £ d stoc hastic matrices, S the multiplicative semigr oup g ener ated by S ; and K the k ernel of S : Let the r ank of the matrices in K be 2 ; and let e be a xed idempotent in K : Assume that n does not con ver g e weakly Then eS 2 n e = f e g ; n 1 ; eS 2 n +1 e = f y g ; n 0 : (2. 65)Proof.Let x 1 ; x 2 be in S : Then the element ex 1 cannot be an idempotent; for if it is, then ex 1 ; being an element in eK ; belongs to Y ; which is a right-zero semigroup, and as such, ( ex 1 ) e = e; contradicting (2. 55). Similarly the element x 2 e cannot be an idempotent; for if it is, then e ( x 2 e ) = e; since x 2 e will then be an element in X ; a left-zero semigroup. Thus, considering the product representation X £ G £ Y of K ; where Y X = f e g and G = f e; y g ; the elements ex 1 and x 2 e; considered as elements in X £ G £ Y ; ha v e representations gi v en by ex 1 = ( x 0 ; y ; y 0 ) ; x 0 2 X ; y 0 2 Y ; x 2 e = ( x 00 ; y ; y 00 ) ; x 00 2 X ; y 00 2 Y : Notice that an element of the form ( x 000 ; e; y 000 ) ; where x 000 2 X and y 000 2 Y ; is an idempo39

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tent since Y X = f e g : Thus, we ha v e ex 1 x 2 e = ( x 0 ; y ; y 0 )( x 00 ; y ; y 00 ) = ( x 0 ; y ( y 0 x 00 ) y ; y 00 ) = ( x 0 ; e; y 00 ) ; which is an idempotent. Since ex 1 x 2 e 2 eK e = G; ex 1 x 2 e = e: This pro v es that eS 2 e = f e g : Let us no w observ e that each element in the set S 2 e is an idempotent, since for an y x 1 ; x 2 in S ; ( x 1 x 2 e )( x 1 x 2 e ) = x 1 x 2 ( ex 1 x 2 e ) = x 1 x 2 e; by the pre vious step. Also, an y element in the set y S is an idempotent, since for an y x 3 2 S ; ( y x 3 )( y x 3 ) = ( y ex 3 )( ey x 3 ) = y ( ex 3 e ) y x 3 = y 3 x 3 = y x 3 ; using (2. 55). Thus, y x 3 = ey x 3 2 Y ; and x 1 x 2 e 2 X : Since Y X = f e g ; it follo ws that for x 1 ; x 2 ; x 3 2 S ; we ha v e ( y x 3 )( x 1 x 2 e ) = e; or ex 3 x 1 x 2 e = y : Thus, we ha v e pro v en: eS 3 e = f y g : No w suppose we ha v e pro v en: eS 2 k e = e for k n: Then since ( S 2 n e )( S 2 n e ) = S 2 n ( eS 2 n e ) = S 2 n e; e v ery element in S 2 n e is an idempotent, and therefore, in X : Also, since ( y S )( y S ) = ( y e ) S ( ey ) S = y ( eS e ) y S = y 3 S = y S ; using (2. 55), e v ery element in y S = ey S is an idempotent, and therefore, in Y : Hence, ( y S )( S 2 n e ) Y X = f e g ; or eS 2 n +1 e = f y g : 40

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Similarly since ( eS 2 )( eS 2 ) = ( eS 2 e ) S 2 = eS 2 ; e v ery element in eS 2 2 Y ; and this means that eS 2 n +2 e = ( eS 2 )( S 2 n e ) Y X = f e g : This pro v es (2. 65) completely No w we will use the equation exx 0 e = e for an y tw o arbitrary elements x; x 0 2 S (see (2. 65)). This gi v es us the equations A ( X 0 21 X 11 + X 0 23 X 31 ) B 0 + AX 0 21 X 13 B = 0 (2. 66) and B ( X 0 31 X 11 + X 0 32 X 21 ) A 0 + B X 0 31 X 12 A = 0 : (2. 67) Ev ery term in (2. 66) and (2. 67) is separately zero since we are dealing with stochastic matrices. Thus, since A and B are strictly positi v e, we ha v e X 0 21 X 13 = 0 and X 0 31 X 12 = 0; (2. 68) AX 0 21 X 11 B 0 = 0 and AX 0 23 X 31 B 0 = 0; (2. 69) B X 0 31 X 11 A 0 and B X 0 32 X 21 A 0 = 0 : (2. 70) Note that in all the equations abo v e the blocks X ij correspond to the element x and the blocks X 0 ij to x 0 ; where x and x 0 are tw o arbitrary elements in S : Let us no w dene the sets T ¤ 1 T 1 and T ¤ 2 T 2 as follo ws: T ¤ 1 = f t 2 T 1 : there e xists x 2 S such that ( X 21 ) j t > 0 for some j 2 C 1 g ; T ¤ 2 = f t 2 T 2 : there e xists x 2 S such that ( X 31 ) j t > 0 for some j 2 C 2 g : Note that T ¤ 1 and T ¤ 2 are dened independent of an y particular element in S : Then it follo ws from (2. 68) that X 13 j T ¤ 1 £ C 2 = 0; X 21 j C 1 £ ( T 1 n T ¤ 1 ) = 0; (2. 71) also, X 12 j T ¤ 2 £ C 1 = 0; X 31 j C 2 £ ( T 2 n T ¤ 2 ) = 0 : (2. 72) 41

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It also follo ws from (2. 69) that X 11 j T ¤ 1 £ ( T 1 [ T 3 ) = 0 ; (2. 73) and from (2. 70) that X 11 j T ¤ 2 £ ( T 2 [ T 3 ) = 0 : (2. 74) No w we ha v e some information about the elements x in S with respect to the partition f C 1 [ T ¤ 2 ; C 2 [ T ¤ 1 g (2. 75) from equations (2. 59), (2. 71), (2. 72), (2. 73), and (2. 74). If e v ery element of S is not c yclic with respect to the partition (2. 75), then we can form a ne w partition from (2. 75) by the addition of ne w subsets of T 1 and T 2 disjoint from T ¤ 1 and T ¤ 2 ; respecti v ely This we do as follo ws: W e dene T ¤¤ 1 and T ¤¤ 2 by T ¤¤ 1 = f t 2 T 1 n T ¤ 1 : there e xists x 2 S such that ( X 11 ) ut > 0 for some u 2 T ¤ 2 g ; T ¤¤ 2 = f t 2 T 2 n T ¤ 2 : there e xists x 2 S such that ( X 11 ) ut > 0 for some u 2 T ¤ 1 g : Then we use (2. 65) ag ain and the equation eS 3 e = f y g : Thus, for arbitrary elements x 00 ; x 0 ; and x in S ; we ha v e X 00 21 X 0 11 X 12 = 0 ; X 00 31 X 0 11 X 13 = 0 ; (2. 76) since y j C 1 £ C 1 and y j C 2 £ C 2 are both zero blocks. It follo ws from (2. 76) that X 13 j T ¤¤ 1 £ C 2 = 0 ; X 12 j T ¤¤ 2 £ C 1 = 0 : (2. 77) From the equation eS 3 e = f y g ; we also ha v e, besides equation (2. 76), the follo wing equations: AX 00 21 X 0 11 X 11 A 0 = 0 ; (2. 78) and B X 00 31 X 0 11 X 11 B 0 = 0 : (2. 79) 42

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Notice that gi v en an y w 2 T ¤¤ 2 and an y t 2 T 2 [ T 3 ; thre e xist v 2 T ¤ 1 ; x 0 2 S m u; x 00 2 S m u; and u 2 C 1 ; such that ( X 00 21 ) uv > 0 ; ( X 0 11 ) v w > 0 ; and A 0ts > 0 for an y s in C 1 : Then (2. 78) implies that X 11 j T ¤¤ 2 £ ( T 2 [ T 3 ) = 0 : (2. 80) Similarly (2. 79) implies that X 11 j T ¤¤ 1 £ ( T 1 [ T 3 ) = 0 : (2. 81) It is no w clear that equations (2. 77), (2. 80), and (2. 81) tak e us closer to the c yclicity partition we ha v e been looking for and this induction process can be continued, if necessary Thus, we no w consider the partition f C 1 [ T ¤ 2 [ T ¤¤ 2 ; C 2 [ T ¤ 1 [ T ¤¤ 1 g (2. 82) If there still remains an element x in S which is not c yclic with respect to the partition (2. 82), we consider (2. 65) ag ain and the equation eS 4 e = f e g to continue the process. As before, for arbitrary elements x 000 ; x 00 ; x 0 ; and x in S ; we obtain the follo wing equations: X 000 21 X 00 11 X 0 11 X 13 = 0; (2. 83) X 000 31 X 00 11 X 0 11 X 12 = 0; (2. 84) AX 000 21 X 00 11 X 0 11 X 11 B 0 = 0; (2. 85) B X 000 31 X 00 11 X 0 11 X 11 A 0 = 0 : (2. 86) From (2. 83), (2. 84), (2. 85), and (2. 86), and from the denitions T ¤¤¤ 1 = f t 2 T 1 n ( T ¤ 1 [ T ¤¤ 1 ) : there e xists x 2 S such that ( X 11 ) ut > 0 for some u 2 T ¤¤ 2 g and T ¤¤¤ 2 = f t 2 T 2 n ( T ¤ 2 [ T ¤¤ 2 ) : there e xists x 2 S such that ( X 11 ) ut > 0 for some u 2 T ¤¤ 1 g ; 43

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it ag ain follo ws as before that X 13 j T ¤¤¤ 1 £ C 2 = 0 ; X 12 j T ¤¤¤ 2 £ C 1 = 0 ; (2. 87) X 11 j T ¤¤¤ 1 £ ( T 1 [ T 3 ) = 0 ; X 11 j T ¤¤¤ 2 £ ( T 2 [ T 3 ) = 0 : (2. 88) W e continue this process if still there remains an element in S which is not c yclic with respect to the partition f C 1 [ T ¤ 2 [ T ¤¤ 2 [ T ¤¤¤ 2 ; C 2 [ T ¤ 1 [ T ¤¤ 1 [ T ¤¤¤ 1 g : (2. 89) This process must terminate after some steps, since both T 1 and T 2 are nite, and we will then ha v e a partition with respect to which all the elements in S are c yclic. Finally let us comment that if there is an idempotent e in K with no zero columns and of rank 2 ; then we will carry out the abo v e analysis using this particular e; and in this case, the analysis is e xtremely simple. If we write e in block form as e = 0@ A 0 0 B 1A ; (2. 90) where A is a c 1 £ c 1 strictly positi v e rank one stochastic matrix inde x ed by C 1 ; B is a c 2 £ c 2 strictly positi v e rank one stochastic matrix inde x ed by C 2 ; c 1 = j C 1 j ; c 2 = j C 2 j ; and c 1 + c 2 = d: Then, the element y is of the form y = 0@ 0 W Z 0 1A ; (2. 91) where W is c 1 £ c 2 ; Z is c 2 £ c 1 ; each ro w of W is the same as an y ro w of B ; and each ro w of Z is the same as an y ro w of A: If we write x 2 S in the same block form as in (2. 90), that is, x = 0@ X 11 X 12 X 21 X 22 1A ; (2. 92) then equation (2. 55), namely that eS e = f y g ; immediately implies that X 11 = 0 ; X 22 = 0 : (2. 93) 44

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This means that S is c yclic with respect to the partition f C 1 ; C 2 g : Thus, we ha v e no w pro v en T H E O R E M 2.6 Let be a pr obability measur e on the Bor el subsets of d £ d stoc hastic matrices suc h that the minimal r ank of the matrices in the closed semigr oup g ener ated by the support of is 2 : Then the sequence of con volution power s n does not con ver g e weakly if f the support of is cyclic. E X A M P L E 7 Consider a tw o-point probability measure on 7 £ 7 stochastic matrices such that S = f x; y g ; where x = 0BBBBBBBBBBBBBBB@ 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 2 1 2 0 0 0 1 2 0 0 0 1 2 0 0 0 0 0 0 1 0 0 1 2 0 0 0 1 2 0 0 0 1 2 0 1 2 0 1CCCCCCCCCCCCCCCA ; y = 0BBBBBBBBBBBBBBB@ 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1CCCCCCCCCCCCCCCA : Note that K consists of rank tw o matrices and K contains an idempotent e gi v en by e = 0BBBBBBBBBBBBBBB@ 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1CCCCCCCCCCCCCCCA : It can be v eried easily that eS e = f y g ; where eK e = f e; y g : Ob viously the sequence n does not con v er ge weakly and S here is c yclic with respect to the partition gi v en by: f C 1 [ T ¤ 2 [ T ¤¤ 2 ; C 2 [ T ¤ 1 g ; 45

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where C 1 = f 6 g ; C 2 = f 7 g ; T 1 = f 1 ; 2 ; 3 g ; T 2 = f 4 ; 5 g ; T ¤ 1 = f 3 g ; T ¤ 2 = f 4 g ; T ¤¤ 2 = f 5 g : W e no w continue with the general d £ d case, still under the assumption of non-weak con v er gence of the sequence n ; b ut this time we consider the case when the common rank of the matrices in the k ernel K is greater than 2 : Consider Example 6 in Section 2.3. W e will use this e xample to produce a probability measure on d £ d stochastic matrices ( d is an y gi v en inte ger > 3 ), where the minimal rank of the matrices in S is 3 ; such that n does not con v er ge weakly and yet, S is not c yclic. First, let A be a 4 £ 4 stochastic matrix. Let A 4 be the last ro w of A: Construct the ( d ¡ 4) £ 4 stochastic matrix A 0 with identical ro ws, each ro w being A 4 = ( a 1 ; a 2 ; a 3 ; a 4 ) Then the d £ d matrix A 00 dened by A 00 = 0@ A 0 A 0 0 1A (2. 94) whose last d ¡ 4 columns are all zero columns, is a stochastic matrix with the same rank as A: If B = ( b ij ) is another 4 £ 4 stochastic matrix, similarly construct B 0 and B 00 = 0@ B 0 B 0 0 1A : Then A 00 B 00 = 0@ AB 0 A 0 B 0 1A : Since A 0 has identical ro ws, so does A 0 B ; and each ro w of A 0 B has the entry 4 X s =1 a s b sj 46

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in its j th column, which is the j th entry in the last ro w of AB : Hence A 0 B = ( AB ) 0 ; and therefore A 00 B 00 = ( AB ) 00 ; which implies that A 7! A 00 is an isomorphism (into). Thus, Example 6 can be recast, through the correspondence A A 00 ; to a similar e xample in the conte xt of d £ d stochastic matrices, d 4 : Expanding on this notion, let us once ag ain start from a 4 £ 4 stochastic matrix A: Append d ¡ 4 zeros to each ro w of A and then add d ¡ 4 identical ro ws of length d; each with zero entries e xcept in the 5 th position, which is 1 ; to create a d £ d stochastic matrix A ¤ of rank 1 more than that of A: That is, A ¤ = 0BBBBBBBBB@ A 0 0 1 0 0 ¢ ¢ ¢ 0 1 0 0 ¢ ¢ ¢ 0 ... 1 0 0 ¢ ¢ ¢ 0 1CCCCCCCCCA : If B is another 4 £ 4 stochastic matrix, similarly construct B ¤ and calculate A ¤ B ¤ = 0BBBBBBBBB@ A 0 0 1 0 0 ¢ ¢ ¢ 0 1 0 0 ¢ ¢ ¢ 0 ... 1 0 0 ¢ ¢ ¢ 0 1CCCCCCCCCA 0BBBBBBBBB@ B 0 0 1 0 0 ¢ ¢ ¢ 0 1 0 0 ¢ ¢ ¢ 0 ... 1 0 0 ¢ ¢ ¢ 0 1CCCCCCCCCA = 0BBBBBBBBB@ AB 0 0 1 0 0 ¢ ¢ ¢ 0 1 0 0 ¢ ¢ ¢ 0 ... 1 0 0 ¢ ¢ ¢ 0 1CCCCCCCCCA to deduce that A 7! A ¤ is an isomorphism (into). Thus, we can ag ain recast Example 6 to 47

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sho w that when the rank of the matrices in K is 4 ; non-weak con v er gence of n may not imply that S is c yclic. More generally we may consider an y rank p; 4 p < d; by constructing the d £ d stochastic matrix A ¤ ( p ) from an y 4 £ 4 stochastic matrix of rank 3 as follo ws: A ¤ ( p ) = 0BBBBBBBBBBBB@ A 0 0 0 I p ¡ 3 0 0 0 0 0 ¢ ¢ ¢ 1 0 0 0 ¢ ¢ ¢ 1 ... 0 0 0 ¢ ¢ ¢ 1 0 1CCCCCCCCCCCCA ; where I p ¡ 3 is the ( p ¡ 3) £ ( p ¡ 3) identity matrix, and where the bottom blocks are of size ( d ¡ p ¡ 1) £ 4 ; ( d ¡ p ¡ 1) £ ( p ¡ 3) ; and ( d ¡ p ¡ 1) £ ( d ¡ p ¡ 1) ; respecti v ely It is not dif cult to see that, once ag ain, A 7! A ¤ ( p ) is an isomorphism. Consequently we conclude that S need not be c yclic in the case of d £ d stochastic matrices when the rank in case is p; 4 p < d: 48

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Chapter 3 W eak Con v er gence in Circulant Matrices 3.1 d £ d Circulant Matrices By a d £ d circulant matrix we mean a matrix of the form 0BBBBBB@ x 0 x 1 ¢ ¢ ¢ x d ¡ 1 x d ¡ 1 x 0 ¢ ¢ ¢ x d ¡ 2 ... ... ... x 1 x 2 ¢ ¢ ¢ x 0 1CCCCCCA where each ro w after the rst is just the pre vious ro w c yclically shifted to the right by one position. Thus, a d £ d matrix x = ( x j k ) ; j ; k = 0 ; 1 ; : : : ; d ¡ 1 ; is a circulant matrix if and only if x j k = x 0 ;k ¡ j ; (3. 1) where the subscripts are tak en modulo d; and the entries x j k are all reals. Clearly a circulant matrix is determined completely by its rst ro w (or column), and we shall, for bre vity denote by cir c ( x 0 ; x 1 ; : : : ; x d ¡ 1 ) the d £ d circulant matrix with rst ro w elements x 0 ; x 1 ; : : : ; x d ¡ 1 : The permutation matrix cir c (0 ; 1 ; 0 ; : : : ; 0) is denoted by P : Circulant matrices are well-studied in the literature (see, for e xample, [D]). The follo wing results are well-kno wn: 1. Let x ¤ denote the conjug ate transpose of the d £ d matrix x: Then the follo wing are equi v alent: i. x is circulant; 49

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ii. x ¤ is circulant; iii. P x = xP ; where the permutation matrix cir c (0 ; 1 ; 0 ; : : : ; 0); i v x = f ( P ) for some polynomial f of de gree less than d: 2. Let x = cir c ( x 0 ; x 1 ; : : : ; x d ¡ 1 ) : Consider the d £ d matrix F whose j th column, 1 j d; is the column with entries 1 p d (1 ; j ¡ 1 ; 2( j ¡ 1) ; : : : ; ( d ¡ 1)( j ¡ 1) ) ; where = exp(2 i=d ) : Dene j = x 0 + x 1 j ¡ 1 + x 2 2( j ¡ 1) + ¢ ¢ ¢ + x d ¡ 1 ( d ¡ 1)( j ¡ 1) : Then the circulant matrix x has the follo wing spectral representation: x = F D x F ¤ ; (3. 2) where F ¤ is the conjug ate transpose of the unitary matrix F ; and D x is the d £ d diagonal matrix diag ( 1 ; 2 ; : : : ; d ) : Note that the unitary matrix F is independent of x: If the rank of x is r ; then e xactly d ¡ r diagonal entries of D x are zeros. Let be a probability measure on the Borel sets of d £ d r eal circulant matrices and let S be the closed (multiplicati v e) semigroup generated by the support S of ; so that S = [ n 1 S n : (3. 3) W e are interested in studying the problem of weak con v er gence of the con v olution sequence ( n ) n 1 ; where, as usual, n +1 ( B ) = Z n f y : y x 2 B g ( dx ) ; for an y Borel set B S : Or equi v alently if X 1 ; X 2 ; : : : are i.i.d. random matrices in S such that P ( X 1 2 B ) = ( B ) ; then P ( X 1 X 2 ¢ ¢ ¢ X n 2 B ) = n ( B ) : The follo wing result is well-kno wn (see [HMu ]). L E M M A 3.1 Assume that the con volution sequence ( n ) n 1 is tight. That is, given > 0 ; we can nd a compact subset K of S suc h that n ( K ) > 1 ¡ for all n 1 : Then, the sequence n ; dened by n = (1 =n ) P nk =1 k ; con ver g es weakly to some idempotent pr obability measur e suc h that = ¤ = ¤ : 50

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The support S is the minimal ideal of S ; and as suc h, consists of all those matrices in S whic h have the minimal r ank. In other wor ds, S = f y 2 S : r ank y r ank x for any x 2 S g : In what follo ws, throughout this paper we assume that the sequence ( n ) is tight so that the assertions in Lemma 3.1 abo v e hold. Since S in (3. 3) is a commutati v e semigroup, the minimal ideal S in Lemma 3.1 is a compact abelian group and is the Haar measure of this group (see [HMu ]). L E M M A 3.2 Let K be a compact abelian gr oup of d £ d matrices of r ank r ; wher e 0 < r d: Then K is topolo gically isomorphic to a compact abelian gr oup of r £ r in vertible matrices with determinant § 1 :Proof.Let e be the identity of K : Then there e xists an in v ertible d £ d matrix y such that y ¡ 1 ey = 0@ 0 0 0 I r 1A ; (3. 4) where I r is the r £ r identity matrix. F or x 2 K ; write y ¡ 1 xy = 0@ A B C D 1A ; (3. 5) where D is r £ r ; A is ( d ¡ r ) £ ( d ¡ r ) ; B is ( d ¡ r ) £ r ; and C is r £ ( d ¡ r ) : Since y ¡ 1 xy = ( y ¡ 1 ey )( y ¡ 1 xy ) = ( y ¡ 1 xy )( y ¡ 1 ey ) ; we ha v e 0@ A B C D 1A = 0@ 0 0 C D 1A = 0@ 0 B 0 D 1A and consequently A = 0 ; B = 0 ; and C = 0 : It is clear that K is topologically isomorphic (under the map x 7! y ¡ 1 xy ) to the group G gi v en by G = 8<: D : y ¡ 1 xy = 0@ 0 0 0 D 1A for some x 2 K 9=; : (3. 6) 51

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Since G is a compact abelian group and the determinant map det : G R is a continuous homomorphism, the range of this map, being a compact group must be either f 1 g or f 1 ; ¡ 1 g : W e will need our ne xt result, the structure of an idempotent d £ d real circulant matrix e = cir c ( e 0 ; e 1 ; : : : ; e d ¡ 1 ) of rank r ; where 0 < r < d; when we discuss the problem of weak con v er gence in circulant matrices especially in the ne xt tw o sections. Recall that e is idempotent if e 2 = e: If 0 ; 1 ; : : : ; d ¡ 1 are the eigen v alues of e; then from (3. 2), e = F ¤ F ¤ ; where ¤ = diag ( 0 ; 1 ; : : : ; d ¡ 1 ) : Thus, the equation e 2 = e translates to F ¤ 2 F ¤ = F ¤ F ¤ ; or diag ( 20 ; 21 ; : : : ; 2d ¡ 1 ) = ¤ 2 = ¤ = diag ( 0 ; 1 ; : : : ; d ¡ 1 ) : In other w ords, j = 0 or 1 ; j = 0 ; 1 ; : : : ; d ¡ 1 : Moreo v er the 0 1 eigen v alues 0 ; 1 ; : : : ; d ¡ 1 satisfy j = d ¡ j ; j = 1 ; 2 ; : : : ; b ( d ¡ 1) = 2 c : Call a sequence a 1 ; a 2 ; : : : ; a d ¡ 1 of real numbers palindr omic if a j = a d ¡ j ; j = 1 ; 2 ; : : : ; b ( d ¡ 1) = 2 c : L E M M A 3.3 Let e = cir c ( e 0 ; e 1 ; : : : ; e d ¡ 1 ) be a d £ d idempotent cir culant matrix with eig en values 0 ; 1 ; : : : ; d ¡ 1 : Then i. eac h j is either 0 or 1; ii. If e has r ank r ; wher e 0 < r < d; then e 0 = r =d; and e xactly r of the j s ar e equal to 1 : In particular if r = d ¡ 1 and d is e ven, ther e ar e two possible idempotents: e = cir c (( d ¡ 1) =d; ¡ 1 =d; ¡ 1 =d; : : : ; ¡ 1 =d ) : (3. 7) 52

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or e = cir c (( d ¡ 1) =d; 1 =d; ¡ 1 =d; 1 =d; : : : ; ¡ 1 =d; 1 =d ) : (3. 8) If r = d ¡ 1 and d is odd, then ther e is only one idempotent and it is given by (3. 7). If r = 1 and d is e ven, ther e ar e two possible idempotents: e = cir c (1 =d; 1 =d; : : : ; 1 =d; 1 =d ) (3. 9) or e = cir c (1 =d; ¡ 1 =d; : : : ; 1 =d; ¡ 1 =d ) : (3. 10) If r = 1 and d is odd, then ther e is only one idempotent and it is given by (3. 9).Proof.i. Clear from the discussion preceeding the lemma. ii. Recall that the rank of a diagonalizable matrix is precisely the number of its nonzero eigen v alues, and that the trace of a matrix is the sum of its eigen v alues. Thus, for the idempotent circulant matrix e; the trace and the rank coincide, leading to the equation de 0 = r : If r = d ¡ 1 and d is e v en, then either 0 = 0 and 1 = 2 = ¢ ¢ ¢ = d ¡ 1 = 1 ; or d= 2 = 0 and the rest of the j s are equal to 1 : (3. 7) and (3. 8) no w follo w from (3. 2).If r = d ¡ 1 and d is odd, then 0 = 0 and 1 = 2 = ¢ ¢ ¢ = d ¡ 1 = 1 ; and the idempotent is uniquely gi v en by (3. 7). If r = 1 and d is e v en, then either 0 = 1 and j = 0 for j = 1 ; 2 ; : : : ; d ¡ 1 ; or d= 2 = 1 and the rest of the j s are zero. Thus, by (3. 2), these tw o possibilities correspond to (3. 9) and (3. 10), respecti v ely If r = 1 and d is odd, then 0 = 1 and j = 0 for j = 1 ; 2 ; : : : ; d ¡ 1 : So there is a unique idempotent, and it is gi v en by (3. 9). 53

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Ne xt, we will characterize the structure of a compact abelian group of d £ d circulant matrices. L E M M A 3.4 Let K be a compact abelian gr oup of d £ d cir culant matrices. Suppose that the r ank of the matrices in K is r ; 0 < r d: Then K is isomorphic to a compact gr oup H of in vertible r by r matrices, wher e eac h matrix in H has the same dia gonal bloc k form as follows: for any x 2 H ; the r st k 1 elements along the dia gonal ar e 1 or ¡ 1 ; and ther e ar e k 2 dia gonal bloc ks, wher e eac h bloc k is a 2 £ 2 r otation matrix of the form 0@ cos sin ¡ sin cos 1A ; wher e k 1 + 2 k 2 = r ; and the number s k 1 and k 2 ar e independent of x 2 H :Proof.W e use the spectral representation (3. 2) of elements in K : F or x 2 K ; write x = F D x F ¤ ; where F and D x are as in (3. 2). Since x 2 K has rank r ; e xactly d ¡ r diagonal entries of D x are zeros. First, we claim that for x 2 K ; if ( D x ) ii = 0 ; then ( D y ) ii = 0 for all y 2 K : In other w ords, the zero entries occur in the same diagonal positions for an y D x ; x 2 K : T o see this, note that if this is not true for some x and y in K ; then since xy = ( F D x F ¤ )( F D y F ¤ ) = F D x D y F ¤ = F D xy F ¤ ; so that D x D y = D xy ; it will follo w that the number of zeros on the diagonal of D xy is strictly greater than that for either D x or D y ; which contradicts that x; y ; and xy ha v e the same rank. Second, if for an y x 2 K ; + i is a nonzero entry on the diagonal of D x ; then since f x k : k 1 g is bounded, j + i j k cannot go to innity or zero, and this means that 54

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j + i j = 1 ; or 2 + 2 = 1 : Thus, each diagonal entry of D x is zero or has absolute v alue 1 : [Notice that if lim k !1 x n k = y ; x; y 2 K ; then if 0 < j + i j < 1 ; where + i is a diagonal entry of x; then D y = lim k !1 D n k x = lim k !1 F ¤ x n k F = F ¤ y F ; so that the number of zeros on the diagonal of D y will e xceed that for D x : This contradicts the f act that x and y ha v e the same rank.] Third, let the j th diagonal entry of both D x and D y ; x; y 2 K ; be nonzero, and be respecti v ely + i and + i : Thus, j = x 0 + x 1 j ¡ 1 + x 2 2( j ¡ 1) + ¢ ¢ ¢ + x d ¡ 1 ( d ¡ 1)( j ¡ 1) = + i ; j 2 ; so that the ( d ¡ j + 2) th diagonal entry of D x is d ¡ j +2 = x 0 + x 1 d ¡ j +1 + x 2 2( d ¡ j +1) + ¢ ¢ ¢ + x d ¡ 1 ( d ¡ 1)( d ¡ j +1) = ¡ i : It is then clear that the ( d ¡ j + 2) th diagonal entry of D y will also be ¡ i ; since k ( d ¡ j +1) is the conjug ate of k ( j ¡ 1) ; for 1 k d ¡ 1 : Since j = d ¡ j +2 ; 1 j d; it is clear that 1 is real, 2 and d are conjug ate, 3 and d ¡ 1 are conjug ate, and so on. It follo ws from the abo v e analysis that we can nd a nite number of permutation matrices P 1 ; P 2 ; : : : ; P m such that for each x 2 K ; the matrix P ¡ 1 m P ¡ 1 m ¡ 1 ¢ ¢ ¢ P ¡ 1 1 D x P 1 P 2 ¢ ¢ ¢ P m 55

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is of the form 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 0 . 0 1 . 1 ¡ 1 . ¡ 1 1 + i 1 0 0 1 ¡ i 1 2 + i 2 0 0 2 ¡ i 2 . 1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA where 2 j + 2 j = 1 for each j ; and the conjug ate eigen v alues are adjacent entries on the diagonal. Notice also that the map 0@ + i 0 0 ¡ i 1A 7! 0@ ¡ 1A is an isomorphism. The lemma is no w clear In the ne xt tw o sections, we consider the special case of 3 £ 3 and 4 £ 4 circulant matrices. 3.2 3 £ 3 Circulant Matrices Let K be the k ernel of the closed semigroup S generated by the support S of a probability measure on 3 £ 3 circulant matrices. As before, we assume that ( n ) is tight, so that K is a compact abelian group. Note that the matrices in K all ha v e the same rank. 56

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W e rst try to determine ho w the matrices in K look lik e. W e discuss separately the cases when the matrices in K ha v e rank 1 ; 2 or 3 : (i) Let the rank of the matrices in K be 1 : Then K is topologically isomorphic to a subgroup of the multiplicati v e group f 1 ; ¡ 1 g : Hence, either K = f e g or K = f e; ¡ e g ; where e = cir c (1 = 3 ; 1 = 3 ; 1 = 3) is the unique idempotent of rank 1 : (ii) Suppose the matrices in K all ha v e rank 2 : If x = cir c ( x 0 ; x 1 ; x 2 ) 2 K ; then the eigen v alues of x are 0 = x 0 + x 1 + x 2 1 = x 0 + x 1 + x 2 2 ; 2 = x 0 + x 1 2 + x 2 ; where = ( ¡ 1 + p 3 i ) = 2 : F or x to ha v e rank 2 ; e xactly one of 0 ; 1 and 2 must be equal to 0 : Note that 1 2 = x 20 + x 21 + x 22 ¡ ( x 0 x 1 + x 0 x 2 + x 1 x 2 ) = (( x 0 ¡ x 1 ) 2 + ( x 0 ¡ x 2 ) 2 + ( x 1 ¡ x 2 ) 2 ) = 2 : Thus, if either 1 = 0 or 2 = 0 ; then x 0 = x 1 = x 2 and consequently x will ha v e rank 1 : Thus, it must be the case that 0 = 0 ; and therefore e v ery matrix in K must be of the form cir c ( x 0 ; x 1 ; ¡ ( x 0 + x 1 )) ; (3. 11) with x 0 and x 1 not both zero. W e sho w that if K is innite then K is topologically isomorphic to the circle group T of comple x numbers z = e i ; 2 [0 ; 2 ) : By Lemma 3.3, e = cir c (2 = 3 ; ¡ 1 = 3 ; ¡ 1 = 3) is the identity of K : From the proof of Lemma 3.2, we kno w that there e xists an in v ertible 3 £ 3 matrix y satisfying (3. 6). In 57

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the present case, we may tak e the matrix y to be the orthogonal matrix y = 0BBB@ 1 = p 3 ¡ 1 = p 6 ¡ 1 = p 2 1 = p 3 ¡ 1 = p 6 1 = p 2 1 = p 3 2 = p 6 0 1CCCA : (3. 12) W ith y gi v en by (3. 12) and x 2 K as in (3. 11), we nd that y ¡ 1 xy = 0BBB@ 0 0 0 0 3 2 x 0 p 3 2 ( x 0 + 2 x 1 ) 0 ¡ p 3 2 ( x 0 + 2 x 1 ) 3 2 x 0 1CCCA : Thus, we conclude that the compact abelian group G of Lemma 3.2, to which K is topologically isomorphic, satises G 8<: 0@ 3 2 x 0 p 3 2 ( x 0 + 2 x 1 ) ¡ p 3 2 ( x 0 + 2 x 1 ) 3 2 x 0 1A : x 0 ; x 1 not both 0 9=; : (3. 13) Note that the determinant of the matrices in G must be § 1 ; since G is compact. The determinant of an element of G is 9 4 x 20 + 3 4 ( x 0 + 2 x 1 ) 2 = 3( x 20 + x 0 x 1 + x 21 ) : Since this quantity is al w ays positi v e, we conclude that all matrices in G ha v e determinant 1 : This leads to the equation x 20 + x 0 x 1 + x 21 = 1 = 3 ; so x 1 = ¡ 3 x 0 § p 12 ¡ 27 x 20 6 (3. 14) with x 0 2 [ ¡ 2 = 3 ; 2 = 3] : In vie w of (3. 11), we deduce that K is the union of the sets f cir c ( x 0 ; f ( x 0 ) ; g ( x 0 )) : x 0 2 [ ¡ 2 = 3 ; 2 = 3] g (3. 15) and f cir c ( x 0 ; g ( x 0 ) ; f ( x 0 )) : x 0 2 [ ¡ 2 = 3 ; 2 = 3] g (3. 16) 58

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where f ( x ) = ¡ 3 x ¡ p 12 ¡ 27 x 2 6 (3. 17) and g ( x ) = ¡ 3 x + p 12 ¡ 27 x 2 6 : (3. 18) Also, (3. 14) implies p 3 2 ( x 0 + 2 x 1 ) = § 1 2 q 4 ¡ 9 x 20 ; (3. 19) and substituting (3. 19) in (3. 13) gi v es a clearer description of the compact abelian group G : G 8<: 0@ 3 2 x 0 1 2 p 4 ¡ 9 x 20 ¡ 1 2 p 4 ¡ 9 x 20 3 2 x 0 1A : x 0 2 [ ¡ 2 = 3 ; 2 = 3] 9=; [ 8<: 0@ 3 2 x 0 ¡ 1 2 p 4 ¡ 9 x 20 1 2 p 4 ¡ 9 x 20 3 2 x 0 1A : x 0 2 [ ¡ 2 = 3 ; 2 = 3] 9=; : No w write x 0 = 2 3 cos : Then by direct computation, one sees that the map e i $ 0@ cos sin ¡ sin cos 1A $ 8<: cir c ( x 0 ; f ( x 0 ) ; g ( x 0 )) if sin 0 cir c ( x 0 ; g ( x 0 ) ; f ( x 0 )) if sin < 0 (3. 20) is a topological isomorphism between the circle group T and K : Here, f and g are gi v en by (3. 17) and (3. 18), respecti v ely (iii) Suppose the matrices in K all ha v e rank 3 : Then K must coincide with the whole of S ; since the identity matrix e = cir c (1 ; 0 ; 0) is in K ; and therefore, S = eS K : This means S itself must be a compact abelian group of 3 £ 3 circulant matrices which ha v e determinant § 1 : Let H = f x = cir c ( x 0 ; x 1 ; x 2 ) 2 S : det x = 1 g : If y 2 S is such that det y = ¡ 1 ; then S = H [ y H : 59

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No w for all x 2 H ; det x = x 30 + x 31 + x 32 ¡ 3 x 0 x 1 x 2 = ( x 0 + x 1 + x 2 )(( x 0 ¡ x 1 ) 2 + ( x 0 ¡ x 2 ) 2 + ( x 1 ¡ x 2 ) 2 ) = 2 = 1 ; (3. 21) so x 0 + x 1 + x 2 > 0 : Moreo v er it is easy to establish (by induction, say) that the entries in each ro w of x n sum to ( x 0 + x 1 + x 2 ) n ; for an y n 1 : Since H is a closed subgroup of the compact group S ; H ; too, is compact. This forces x 0 + x 1 + x 2 = 1 ; (3. 22) so e v ery element of H is of the form cir c ( x 0 ; x 1 ; 1 ¡ ( x 0 + x 1 )) : (3. 23) (3. 21) and (3. 22) then imply ( x 0 ¡ x 1 ) 2 + (2 x 0 + x 1 ¡ 1) 2 + (2 x 1 + x 0 ¡ 1) 2 = 2 ; or x 20 + x 21 + x 0 x 1 = x 0 + x 1 : (3. 24) Thus, x 1 = 1 ¡ x § p (1 + 3 x 0 )(1 ¡ x 0 ) 2 (3. 25) with x 0 2 [ ¡ 1 = 3 ; 1] : In vie w of (3. 23), we conclude that H is the union of the sets f cir c ( x 0 ; f ( x 0 ) ; g ( x 0 )) : x 0 2 [ ¡ 1 = 3 ; 1] g and f cir c ( x 0 ; g ( x 0 ) ; f ( x 0 )) : x 0 2 [ ¡ 1 = 3 ; 1] g where f ( x ) = 1 ¡ x ¡ p (1 + 3 x )(1 ¡ x ) 2 (3. 26) 60

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and g ( x ) = 1 ¡ x + p (1 + 3 x )(1 ¡ x ) 2 : (3. 27) Consider no w the orthogonal matrix s gi v en by s = 0BBB@ 1 = p 6 1 = p 2 1 = p 3 1 = p 6 ¡ 1 = p 2 1 = p 3 ¡ 2 = p 6 0 1 = p 3 1CCCA : F or x = cir c ( x 0 ; x 1 ; 1 ¡ ( x 0 + x 1 )) 2 H ; we ha v e, using (3. 25), s ¡ 1 xs = 0BBB@ 1 2 (3 x 0 ¡ 1) § p 3 2 p (1 + 3 x 0 )(1 ¡ x 0 ) 0 ¨ p 3 2 p (1 + 3 x 0 )(1 ¡ x 0 ) 1 2 (3 x 0 ¡ 1) 0 0 0 1 1CCCA : Thus, writing x 0 = (2 cos + 1) = 3 ; direct computation yields the topological isomor phism e i $ 0BBB@ cos sin 0 ¡ sin cos 0 0 0 1 1CCCA $ 8<: cir c ( x 0 ; f ( x 0 ) ; g ( x 0 )) if sin 0 cir c ( x 0 ; g ( x 0 ) ; f ( x 0 )) if sin < 0 (3. 28) between the circle group and H : Here, f and g are gi v en by (3. 26) and (3. 27), respecti v ely It follo ws that S is topologically isomorphic either to the circle group T or to the direct product of T and f 1 ; ¡ 1 g : Ne xt, we gi v e a well-kno wn characterization of the proper closed (compact) subgroups of the circle group T : L E M M A 3.5 Every pr oper closed subgr oup of the cir cle gr oup T is nite cyclic.Proof.Observ e that T may be identied with U = [0 ; 1) under addition mod 1 with its usual topology at e v ery point, e xcept that an open neighborhood of 0 is [0 ; a ) [ ( b; 1) ; where a; b 2 (0 ; 1) : 61

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Let H be a proper compact subgroup of U : Then the identity 0 2 H ; and we claim that 0 and e v ery point in H must be isolated. If 0 is not isolated, then gi v en > 0 ; there must be x 2 H such that x 2 (0 ; ) : No w gi v en an y y 2 U ; there e xists an inte ger n such that y nx y + 2 ": Since nx 2 H ; this means that H is dense in U : It follo ws by compactness of H that H = U : Therefore, H cannot be proper W e conclude that the points of a proper compact subgroup H are all isolated, so that such a subgroup must be nite. Let H = f h 0 ; h 1 ; : : : ; h n ¡ 1 g ; where 0 = h 0 < h 1 < ¢ ¢ ¢ < h n ¡ 1 and n is the cardinality of H : Thus h 1 is the smallest positi v e element in H : W e claim that h k = k h 1 for all k = 1 ; 2 ; : : : ; n ¡ 1 : If this is not the case, let s > 1 be the rst inte ger such that h s 6 = sh 1 : Since ( s ¡ 1) h 1 = h s ¡ 1 < h s and h s is the smallest element in H greater than h s ¡ 1 ; we ha v e ( s ¡ 1) h 1 < h s < sh 1 : But this implies 0 < h s ¡ ( s ¡ 1) h 1 < sh 1 ¡ ( s ¡ 1) h 1 = h 1 : (3. 29) Since h s ¡ ( s ¡ 1) h 1 2 H ; (3. 29) is a contradiction to the minimality of h 1 : W e conclude that H is nite c yclic, as desired. Before we state the main result of this section, we state without proof a well-kno wn necessary and suf cient condition for the con v er gence of ( n ) ; under the condition of tightness. The v ersion we present here is tak en from part (iii) of Theorem 2.13 in [HMu ], and is adapted to our present situation. L E M M A 3.6 Let be a pr obability measur e on a commutative semigr oup S of d £ d r eal matrices suc h that the support S of g ener ates S : Suppose that the sequence ( n ) n 1 is tight. Then K ; the k ernel (that is, the smallest ideal) of S ; is a compact abelian gr oup. Furthermor e ( n ) con ver g es weakly if and only if ther e does not e xist a closed subgr oup H of the compact abelian gr oup K suc h that eS g H (3. 30) 62

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for some g 2 K n H : Her e e is the identity of K : In vie w of the abo v e discussion and Lemma 3.6, we ha v e the follo wing T H E O R E M 3.1 Let be a pr obability measur e on 3 £ 3 r eal cir culant matrices. Let S be the closed commutative semigr oup g ener ated by the support S of ; and let K be the k ernel of S : Suppose ( n ) n 1 is tight. Then (i) If K consists of r ank 1 matrices, then either (a) K = f e g and n w e ; or (b) K = f e; ¡ e g and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; x 1 ; ¡ 1 ¡ x 0 ¡ x 1 ) : x 0 ; x 1 2 R g : Her e e = cir c (1 = 3 ; 1 = 3 ; 1 = 3) and e is the point mass at e: (ii) If K consists of r ank 2 matrices, then either (a) K = f e g and n w e ; (b) K = f e; ¡ e g and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; x 0 + 1 ; x 0 + 1) : x 0 2 R g ; or (c) K is topolo gically isomorphic to the cir cle gr oup T (or a nite subgr oup T 0 of T ) and ( n ) con ver g es weakly if and only if the ima g e of S under the topolo gical isomorphism given by (3. 20) is not contained in a coset of a nite subgr oup of T (r espectively a pr oper subgr oup of T 0 ). Her e e = cir c (2 = 3 ; ¡ 1 = 3 ; ¡ 1 = 3) : (iii) If K consists of r ank 3 matrices, then either 63

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(a) K = f I 3 g and n w I 3 ; (b) K = f I 3 ; ¡ I 3 g and ( n ) con ver g es weakly if and only if S 6 = f¡ I 3 g ; or (c) K = S is topolo gically isomorphic to the cir cle gr oup T or to T £ f 1 ; ¡ 1 g (or a nite subgr oup T 0 of T or T £ f 1 ; ¡ 1 g ) and ( n ) con ver g es weakly if and only if the ima g e of S under the topolo gical isomorphism given by (3. 28) is not contained in a coset of a nite subgr oup of T or T £ f 1 ; ¡ 1 g (r espectively a pr oper subgr oup of T 0 ). Let us consider tw o e xamples. E X A M P L E 8 Suppose that S f cir c ( x; x ¡ 1 ; x ¡ 1) : x 2 R g : Suppose also that for some positi v e inte ger n; n ( e ) > 0 ; where e is the idempotent circulant matrix cir c (2 = 3 ; ¡ 1 = 3 ; ¡ 1 = 3) : It is easily v eried that for an y matrix y 2 S ; y e = ey = e: In other w ords, the k ernel K of S = [ k 1 S k is f e g : By Lemma 2.19 (page 106 in [HMu]), it follo ws that lim n !1 n ( e ) = 1 so that n con v er ges weakly to e : E X A M P L E 9 Suppose that S f cir c ( x 0 ; x 1 ; x 2 ) : x 20 + x 21 + x 22 ¡ x 0 x 1 ¡ x 1 x 2 ¡ x 0 x 2 = 1 g : 64

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Let us denote the set on the right by M : Suppose also that for some positi v e inte ger n 0 ; n 0 ( A ) > 0 ; where the set A is gi v en by A = ( cir c x 0 ; ¡ 1 2 x 0 + p 3 2 r 4 9 ¡ x 20 ; ¡ 1 2 x 0 ¡ p 3 2 r 4 9 ¡ x 20 : x 20 4 9 ) : It can also be v eried that M is a semigroup. Indeed, if ( x 0 ¡ x 1 ) 2 + ( x 1 ¡ x 2 ) 2 + ( x 0 ¡ x 2 ) 2 = 2 and ( y 0 ¡ y 1 ) 2 + ( y 1 ¡ y 2 ) 2 + ( y 0 ¡ y 2 ) 2 = 2 ; then cir c ( x 0 ; x 1 ; x 2 ) ¢ cir c ( y 0 ; y 1 ; y 2 ) = cir c ( z 0 ; z 1 ; z 2 ) ; and ( z 0 ¡ z 1 ) 2 + ( z 1 ¡ z 2 ) 2 + ( z 0 ¡ z 2 ) 2 = [ x 0 ( y 0 ¡ y 1 ) + x 1 ( y 2 ¡ y 0 ) + x 2 ( y 1 ¡ y 2 )] 2 + [ x 0 ( y 1 ¡ y 2 ) + x 1 ( y 0 ¡ y 1 ) + x 2 ( y 2 ¡ y 0 )] 2 + [ x 0 ( y 0 ¡ y 2 ) + x 1 ( y 2 ¡ y 1 ) + x 2 ( y 1 ¡ y 0 )] 2 = 2( x 20 + x 21 + x 22 ) ¡ 2( x 0 x 1 + x 0 x 2 + x 1 x 2 )( y 2 0 + y 2 1 + y 2 2 ¡ y 0 y 1 ¡ y 1 y 2 ¡ y 0 y 2 ) = 2 : Also, notice that y 2 M and det y = 0 imply that y 2 A; and all matrices in A ha v e rank 2 : It follo ws that A \ S ; where S = [ n 1 S n ; is a compact abelian group, and the k ernel of S : By Proposition 2.19 (page 106 in [HMu ]), it also follo ws lim n !1 n ( A ) = 1 65

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so that ( n ) is tight. It is also easily seen that if y is the orthogonal matrix gi v en by y = 0BBB@ 1 = p 3 1 = p 6 ¡ 1 = p 2 1 = p 3 1 = p 6 1 = p 2 1 = p 3 ¡ 2 = p 6 0 1CCCA ; then y ¡ 1 Ay is the circle group. R E M A R K 1 (Personal communication from Karl H. Hofmann of Darmstadt, German y) An y closed innite abelian subgroup C of S O (3) is a circle group. The reason is, briey the follo wing: C is a Lie group, and the identity component C 0 of C must, thus, be open in C : It follo ws that C 0 is a torus. In S O (3) ; ho we v er each torus is a circle group, and is maximal abelian, implying that C is the circle group. In the case of 4 £ 4 circulant matrices (considered ne xt), we will observ e the same type of beha vior with the matrices in the k ernel group K : 3.3 4 £ 4 Circulant Matrices Let K be the k ernel of the closed semigroup S generated by the support S of a probability measure on 4 £ 4 circulant matrices. As before, we assume that ( n ) is tight, so that K is a compact abelian group. W e rst try to determine ho w the matrices in K look lik e. Note that the eigen v alues of x = cir c ( x 0 ; x 1 ; x 2 ; x 3 ) 2 K are gi v en by 0 = x 0 + x 1 + x 2 + x 3 (3. 31) 1 = ( x 0 ¡ x 2 ) + ( x 1 ¡ x 3 ) i; (3. 32) 2 = x 0 ¡ x 1 + x 2 ¡ x 3 ; (3. 33) 3 = 1 = ( x 0 ¡ x 2 ) ¡ ( x 1 ¡ x 3 ) i: (3. 34) 66

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Also, if F is the 4 £ 4 F ourier matrix of section 3.1, then we ha v e the spectral representation (see (3. 2)) x = F diag ( 0 ; 1 ; 2 ; 3 ) F ¤ : (3. 35) F or notational con v enience, we will denote by D x the diagonal matrix on the right side of (3. 35). Note from (3. 35) that the rank of x coincides with that of D x ; so it is e xactly the number of nonzero j : It is also clear from (3. 35) that for e v ery k 1 ; x k = F D k x F ¤ : No w let j 6 = 0 : Then since ( x k ) k 1 is bounded, j j j k cannot go to innity as k 1 : Thus, j j j 1 : Suppose j j j < 1 : T ak e a subsequence ( x n k ) of ( x k ) con v er ging to y = F D y F ¤ 2 K : Then y = lim k !1 x n k = lim k !1 F D n k x F ¤ = F lim k !1 D n k x F ¤ ; so D y = lim k !1 D n k x : Since j j j < 1 ; the ( j ; j ) entry of D y must be zero. This means D y has rank strictly less than that of x; contradicting y 2 K : W e conclude that if j 6 = 0 then j j j = 1 : No w we discuss separately the cases when the matrices in K ha v e rank 1 ; 2 ; 3 or 4 : T o f acilitate the discussion, we will mak e use of the v ector z = ( 0 ; 2 ; j 1 j 2 ) = ( x 0 + x 1 + x 2 + x 3 ; x 0 ¡ x 1 + x 2 ¡ x 3 ; ( x 0 ¡ x 2 ) 2 + ( x 1 ¡ x 3 ) 2 ) : (i) Let the rank of the matrices in K be 1 : Then either z = ( § 1 ; 0 ; 0) or z = (0 ; § 1 ; 0) : In either case we ha v e ( x 0 ¡ x 2 ) 2 + ( x 1 ¡ x 3 ) 2 = 0 ; so x 0 = x 2 ; x 1 = x 3 : 67

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If z = ( § 1 ; 0 ; 0) ; then we ha v e x 0 + x 1 + x 2 + x 3 = 2( x 0 + x 1 ) = § 1 ; x 0 ¡ x 1 + x 2 ¡ x 3 = 2( x 0 ¡ x 1 ) = 0 : Hence § 1 = 2 = x 0 + x 1 ; x 0 = x 1 ; so x 0 = § 1 = 4 : This gi v es the circulants e 11 = cir c (1 = 4 ; 1 = 4 ; 1 = 4 ; 1 = 4) ; ¡ e 11 = cir c ( ¡ 1 = 4 ; ¡ 1 = 4 ; ¡ 1 = 4 ; ¡ 1 = 4) : On the other hand, if z = (0 ; § 1 ; 0) ; then we get § 1 = 2 = x 0 ¡ x 1 ; x 0 = ¡ x 1 ; which gi v e the circulants e 12 = cir c (1 = 4 ; ¡ 1 = 4 ; 1 = 4 ; ¡ 1 = 4) ; ¡ e 12 = cir c ( ¡ 1 = 4 ; 1 = 4 ; ¡ 1 = 4 ; 1 = 4) : Observ e that e 11 and e 12 are idempotent rank 1 circulants. W e conclude that either K = f e 11 g or K = f e 11 ; ¡ e 11 g ; or K = f e 12 g or K = f e 12 ; ¡ e 12 g : (ii) Suppose the matrices in K all ha v e rank 2 : Then either z = ( § 1 ; § 1 ; 0) or z = (0 ; 0 ; 1) : The rst possibility forces x 0 = x 2 ; x 1 = x 3 : When z = (1 ; 1 ; 0) ; we ha v e the equations 2( x 0 + x 1 ) = 1 ; 2( x 0 ¡ x 1 ) = 1 ; so x 0 = 1 = 2 and x 1 = 0 : This gi v es the idempotent rank 2 circulant e 21 = cir c (1 = 2 ; 0 ; 1 = 2 ; 0) : Similarly when z = ( ¡ 1 ; ¡ 1 ; 0) ; we get the circulant ¡ e 21 = cir c ( ¡ 1 = 2 ; 0 ; ¡ 1 = 2 ; 0) : 68

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When z = (1 ; ¡ 1 ; 0) we ha v e f = cir c (0 ; 1 = 2 ; 0 ; 1 = 2) ; and when z = ( ¡ 1 ; 1 ; 0) we ha v e ¡ f = cir c (0 ; ¡ 1 = 2 ; 0 ; ¡ 1 = 2) : A quick calculation tells us that f e 21 ; ¡ e 21 ; f ; ¡ f g is isomorphic to the direct product of the multiplicati v e group f 1 ; ¡ 1 g with itself. Thus, in the rst possibility when z = ( § 1 ; § 1 ; 0) ; we ha v e that K is a subgroup of f e 21 ; ¡ e 21 ; f ; ¡ f g : The second possibility when z = (0 ; 0 ; 1) ; leads to x 0 + x 2 = x 1 + x 3 = 0 and ( x 0 ¡ x 2 ) 2 + ( x 1 ¡ x 3 ) 2 = 1 ; so (2 x 0 ) 2 + (2 x 1 ) 2 = 1 ; so x 1 = § q 1 = 4 ¡ x 20 : This means K consists of circulants of the form x = cir c ( x 0 ; x 1 ; ¡ x 0 ; ¡ x 1 ) = cir c x 0 ; § q 1 = 4 ¡ x 20 ; ¡ x 0 ; ¨ q 1 = 4 ¡ x 20 ; x 0 2 [ ¡ 1 = 2 ; 1 = 2] : Note that if D = 1 2 0BBBBBB@ 1 1 ¡ 1 ¡ 1 1 ¡ 1 ¡ 1 1 1 ¡ 1 1 ¡ 1 1 1 1 1 1CCCCCCA ; (3. 36) 69

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then D is orthogonal and D xD ¡ 1 = 0BBBBBB@ 2 x 0 ¡ 2 x 1 0 0 2 x 1 2 x 0 0 0 0 0 0 0 0 0 0 0 1CCCCCCA = 0BBBBBB@ cos ¡ sin 0 0 sin cos 0 0 0 0 0 0 0 0 0 0 1CCCCCCA where x 0 = 1 2 cos and x 0 2 [ ¡ 1 = 2 ; 1 = 2] : Thus, in this case K = cir c x 0 ; § q 1 = 4 ¡ x 20 ; ¡ x 0 ; ¨ q 1 = 4 ¡ x 20 : x 0 2 [ ¡ 1 = 2 ; 1 = 2] is a compact group topologically isomorphic to the circle group. Observ e that the identity of K is e 22 = cir c (1 = 2 ; 0 ; ¡ 1 = 2 ; 0) ; the other idempotent rank 2 circulant. (iii) Suppose the matrices in K all ha v e rank 3 : Then either z = ( § 1 ; 0 ; 1) or z = (0 ; § 1 ; 1) : When z = (1 ; 0 ; 1) ; we get the equations 1 = 2 = x 0 + x 2 = x 1 + x 3 ; and (2 x 0 ¡ 1 = 2) 2 + (2 x 1 ¡ 1 = 2) 2 = 1 : Thus, x 1 = 1 4 § 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; and we deduce that K consists of circulants of the form x = cir c ( x 0 ; x 1 ; 1 = 2 ¡ x 0 ; 1 = 2 ¡ x 1 ) = cir c x 0 ; 1 4 § 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; 1 = 2 ¡ x 0 ; 1 4 ¨ 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) 70

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with x 0 2 [ ¡ 1 = 4 ; 3 = 4] : Let D 1 be the orthogonal matrix D 1 = 1 2 0BBBBBB@ 1 1 ¡ 1 ¡ 1 1 ¡ 1 ¡ 1 1 1 1 1 1 1 ¡ 1 1 ¡ 1 1CCCCCCA : (3. 37) Then D 1 xD ¡ 1 1 = 0BBBBBB@ 2 x 0 ¡ 1 = 2 ¡ (2 x 1 ¡ 1 = 2) 0 0 2 x 1 ¡ 1 = 2 2 x 0 ¡ 1 = 2 0 0 0 0 1 0 0 0 0 0 1CCCCCCA = 0BBBBBB@ cos ¡ sin 0 0 sin cos 0 0 0 0 1 0 0 0 0 0 1CCCCCCA where x 0 = 1 4 + 1 2 cos and x 0 2 [ ¡ 1 = 4 ; 3 = 4] : Hence, in this case, K = cir c x 0 ; 1 4 § 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; 1 = 2 ¡ x 0 ; 1 4 ¨ 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; with x 0 2 [ ¡ 1 = 4 ; 3 = 4] ; is a compact group topologically isomorphic to the circle group. Note that the identity of K is e 31 = cir c (3 = 4 ; 1 = 4 ; ¡ 1 = 4 ; 1 = 4) ; one of tw o idempotent rank 3 circulants. A similar situation happens when z = (0 ; 1 ; 1) : W e get the equations 1 = 2 = x 0 + x 2 = ¡ ( x 1 + x 3 ) ; and (2 x 0 ¡ 1 = 2) 2 + (2 x 1 + 1 = 2) 2 = 1 ; so x 1 = ¡ 1 4 § 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) 71

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and K therefore consists of circulants of the form x = cir c ( x 0 ; x 1 ; 1 = 2 ¡ x 0 ; ¡ 1 = 2 ¡ x 1 ) = cir c x 0 ; ¡ 1 4 § 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; 1 = 2 ¡ x 0 ; ¡ 1 4 ¨ 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) with x 0 2 [ ¡ 1 = 4 ; 3 = 4] : Let D be the orthogonal matrix gi v en by (3. 36). Then D xD ¡ 1 = 0BBBBBB@ 2 x 0 ¡ 1 = 2 ¡ (2 x 1 + 1 = 2) 0 0 2 x 1 + 1 = 2 2 x 0 ¡ 1 = 2 0 0 0 0 1 0 0 0 0 0 1CCCCCCA = 0BBBBBB@ cos ¡ sin 0 0 sin cos 0 0 0 0 1 0 0 0 0 0 1CCCCCCA where x 0 = 1 4 + 1 2 cos and x 0 2 [ ¡ 1 = 4 ; 3 = 4] : Ag ain, in this case K = cir c x 0 ; ¡ 1 4 § 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; 1 = 2 ¡ x 0 ; ¡ 1 4 ¨ 1 2 p (3 ¡ 4 x 0 )(1 + 4 x 0 ) ; with x 0 2 [ ¡ 1 = 4 ; 3 = 4] ; is a compact group topologically isomorphic to the circle group. The identity of K is e 32 = cir c (3 = 4 ; ¡ 1 = 4 ; ¡ 1 = 4 ; ¡ 1 = 4) ; the other idempotent rank 3 circulant. No w note that when z = ( ¡ 1 ; 0 ; 1) we get the equations ¡ 1 = 2 = x 0 + x 2 = x 1 + x 3 ; and (2 x 0 + 1 = 2) 2 + (2 x 1 + 1 = 2) 2 = 1 : If K is to e xist in this case, then its identity must ha v e x 0 = 3 = 4 : The last equation, ho we v er leads to 4 + (2 x 1 + 1 = 2) 2 = 1 ; 72

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which does not ha v e a real solution for x 1 : W e conclude that we cannot ha v e z = ( ¡ 1 ; 0 ; 1) : The same ar gument may be used to sho w the none xistence of K when z = (0 ; ¡ 1 ; 1) : (i v) Suppose the matrices in K all ha v e rank 4 : Then z = ( § 1 ; § 1 ; 1) : When z = (1 ; 1 ; 1) ; we get the equations 1 = x 0 + x 2 ; x 3 = ¡ x 1 ; 1 = (2 x 0 ¡ 1) 2 + (2 x 1 ) 2 : Thus, x 1 = § p x 0 (1 ¡ x 0 ) : Therefore, K consists of circulants of the form x = cir c ( x 0 ; x 1 ; 1 ¡ x 0 ; ¡ x 1 ) = cir c x 0 ; § p x 0 (1 ¡ x 0 ) ; 1 ¡ x 0 ; ¨ p x 0 (1 ¡ x 0 ) ; x 0 2 [0 ; 1] : Let D be the orthogonal matrix gi v en by (3. 36). Then D xD ¡ 1 = 0BBBBBB@ 2 x 0 ¡ 1 ¡ 2 x 1 0 0 2 x 1 2 x 0 ¡ 1 0 0 0 0 1 0 0 0 0 1 1CCCCCCA = 0BBBBBB@ cos ¡ sin 0 0 sin cos 0 0 0 0 1 0 0 0 0 1 1CCCCCCA where x 0 = 1+cos 2 = cos 2 ( = 2) and x 0 2 [0 ; 1] : Once ag ain, in this case K = cir c x 0 ; § p x 0 (1 ¡ x 0 ) ; 1 ¡ x 0 ; ¨ p x 0 (1 ¡ x 0 ) : x 0 2 [0 ; 1] is a compact group topologically isomorphic to the circle group. The identity of K is I 4 = cir c (1 ; 0 ; 0 ; 0) : An ar gument similar to the one gi v en abo v e for the rank 3 case may be used to deduce that the cases when z is equal to either (1 ; ¡ 1 ; 1) ; ( ¡ 1 ; 1 ; 1) or ( ¡ 1 ; ¡ 1 ; 1) are not possible. 73

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The preceding discussion leads to the follo wing T H E O R E M 3.2 Let be a pr obability measur e on 4 £ 4 r eal cir culant matrices. Let S be the closed commutative semigr oup g ener ated by the support S of ; and let K be the k ernel of S : Suppose ( n ) n 1 is tight. Then (i) If K consists of r ank 1 matrices, then either (a) K = f e 11 g and n w e 11 ; or (b) K = f e 11 ; ¡ e 11 g and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; x 1 ; x 2 ; ¡ 1 ¡ x 0 ¡ x 1 ¡ x 2 ) : x 0 ; x 1 ; x 2 2 R g ; or (c) K = f e 2 g and n w e 12 ; or (d) K = f e 12 ; ¡ e 12 g and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; x 1 ; x 2 ; 1 + x 0 ¡ x 1 + x 2 ) : x 0 ; x 1 ; x 2 2 R g Her e e 11 = cir c (1 = 4 ; 1 = 4 ; 1 = 4 ; 1 = 4) and e 12 = cir c (1 = 4 ; ¡ 1 = 4 ; 1 = 4 ; ¡ 1 = 4) : x is the point mass at x: (ii) If K consists of r ank 2 matrices, then either (a) K = f e 21 g and n w e 21 ; or (b) K is isomorphic to a nontrivial subgr oup of f 1 ; ¡ 1 g 2 ; and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; x 1 ; ¡ 1 ¡ x 0 ; ¡ x 1 ) : x 0 ; x 1 2 R g [ f cir c ( x 0 ; x 1 ; ¡ x 0 ; § 1 ¡ x 1 ) : x 0 ; x 1 2 R g ; or (c) K is topolo gically isomorphic to the cir cle gr oup T (or a nite subgr oup T 0 of T ) and ( n ) con ver g es weakly if and only if e 22 S is not contained in a pr oper coset of a nite subgr oup of K : 74

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Her e e 21 = cir c (1 = 2 ; 0 ; 1 = 2 ; 0) and e 22 = cir c (1 = 2 ; 0 ; ¡ 1 = 2 ; 0) : (iii) If K consists of r ank 3 matrices, then either (a) K = f e 31 g and n w e 31 ; or (b) K = f e 31 ; ¡ e 31 g and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; ¡ 1 ¡ x 0 ; 1 + x 0 ; ¡ 1 ¡ x 0 ) : x 0 2 R g ; or (c) K = f e 32 g and n w e 32 ; or (d) K = f e 32 ; ¡ e 32 g and ( n ) con ver g es weakly if and only if S 6 f cir c ( x 0 ; 1 + x 0 ; 1 + x 0 ; 1 + x 0 ) : x 0 2 R g ; or (e) K is topolo gically isomorphic to the cir cle gr oup T (or a nite subgr oup T 0 of T ) and ( n ) con ver g es weakly if and only if neither e 31 S nor e 32 S is contained in a pr oper coset of a nite subgr oup of K : Her e e 31 = cir c (3 = 4 ; 1 = 4 ; ¡ 1 = 4 ; 1 = 4) and e 32 = cir c (3 = 4 ; ¡ 1 = 4 ; ¡ 1 = 4 ; ¡ 1 = 4) : (iv) If K consists of r ank 4 matrices, then either (a) K = f I 4 g and n w I 4 ; or (b) K = f I 4 ; ¡ I 4 g and ( n ) con ver g es weakly if and only if S 6 = f¡ I 4 g ; or (c) K = S topolo gically isomorphic to the cir cle gr oup T (or a nite subgr oup T 0 of T ) and ( n ) con ver g es weakly if and only if S is not contained in a pr oper coset of a nite subgr oup of S : 75

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3.4 d £ d T oeplitz Matrices In this section, we e xtend the in v estig ation of the weak con v er gence problem to a wider class of certain T oeplitz matrices, which are matrices whose elements along a diagonal are constant. A d £ d T oeplitz matrix x = ( x ij ) ; i; j = 0 ; 1 ; : : : ; d ¡ 1 ; is completely determined by 2 d ¡ 1 constants x k ; k = 0 ; § 1 ; § 2 ; : : : ; § ( d ¡ 1) satisfying x ij = x j ¡ i : Thus, if we inde x the diagonals of x; from the lo wer left to the upper right, using the inte gers from ¡ ( d ¡ 1) to d ¡ 1 ; then the common v alue of the entries in the k th diagonal is precisely x k ; k = ¡ ( d ¡ 1) ; : : : ; d ¡ 1 : Here, we consider the class S d ( t ) of d £ d T oeplitz matrices satisfying x ¡ k = x d ¡ k t ( k = 1 ; 2 ; : : : ; d ¡ 1) where t is a x ed parameter That is, S d ( t ) = 8>>>>>>>>>>>><>>>>>>>>>>>>: 0BBBBBBBBBBBB@ x 0 x 1 x 2 ¢ ¢ ¢ x d ¡ 2 x d ¡ 1 x d ¡ 1 t x 0 x 1 ¢ ¢ ¢ x d ¡ 3 x d ¡ 2 x d ¡ 2 t x d ¡ 1 t x 0 ¢ ¢ ¢ x d ¡ 4 x d ¡ 3 ... ... ... . ... ... x 2 t x 3 t x 4 t ¢ ¢ ¢ x 0 x 1 x 1 t x 2 t x 3 t ¢ ¢ ¢ x d ¡ 1 t x 0 1CCCCCCCCCCCCA : x 0 ; x 1 ; : : : ; x d ¡ 1 2 R 9>>>>>>>>>>>>=>>>>>>>>>>>>; : Note that S d (1) is the class of circulant matrices. Note also that S d ( t ) is a commutati v e semigroup under ordinary matrix multiplication. Ne xt we discuss the spectral representation of a matrix x in S d ( t ) : F or bre vity we shall identify x using its rst ro w elements, and write x = toep ( x 0 ; x 1 ; : : : ; x d ¡ 1 ) 76

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to mean x = 0BBBBBBBBBBBB@ x 0 x 1 x 2 ¢ ¢ ¢ x d ¡ 2 x d ¡ 1 x d ¡ 1 t x 0 x 1 ¢ ¢ ¢ x d ¡ 3 x d ¡ 2 x d ¡ 2 t x d ¡ 1 t x 0 ¢ ¢ ¢ x d ¡ 4 x d ¡ 3 ... ... ... . ... ... x 2 t x 3 t x 4 t ¢ ¢ ¢ x 0 x 1 x 1 t x 2 t x 3 t ¢ ¢ ¢ x d ¡ 1 t x 0 1CCCCCCCCCCCCA : Consider the d £ d matrix P whose j th column, 1 j d; is the column with entries (1 ; s! j ¡ 1 ; s 2 2( j ¡ 1) ; : : : ; s d ¡ 1 ( d ¡ 1)( j ¡ 1) ) ; (3. 38) where = exp(2 i=d ) and s = d p t: Observ e that the columns of P are linearly independent. No w dene j = d ¡ 1 X k =0 x k s k k ( j ¡ 1) ( j = 1 ; 2 ; : : : ; d ) : (3. 39) Then x has the follo wing spectral representation: x = P D x P ¡ 1 ; (3. 40) where D x is the d £ d diagonal matrix diag ( 1 ; 2 ; : : : ; d ) : In what follo ws, we assume that the sequence ( n ) ; where is a probability measure on the Borel sets of S d ( t ) ; is tight, so that the assertions in Lemma 3.1 ag ain hold. If S is the support of and S = [ n 1 S n ; (3. 41) then, as before, S is a commutati v e semigroup, the minimal ideal S in Lemma 3.1 is a compact abelian group, is the Haar measure of this group, and, as before, we ha v e Lemma 3.2. W e will also need L E M M A 3.7 Let e = toep ( e 0 ; e 1 ; : : : ; e d ¡ 1 ) be a d £ d idempotent matrix in S d ( t ) with eig en values 0 ; 1 ; : : : ; d ¡ 1 : Write s = d p t: Then 77

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i. eac h j is either 0 or 1; ii. If e has r ank r ; wher e 0 < r < d; then e 0 = r =d; and e xactly r of the j s ar e equal to 1 : In particular if r = d ¡ 1 and d is e ven, ther e ar e two possible idempotents: e = toep (( d ¡ 1) =d; ¡ s ¡ 1 =d; ¡ s ¡ 2 =d; : : : ; ¡ s ¡ ( d ¡ 1) =d ) : (3. 42) or e = toep (( d ¡ 1) =d; s ¡ 1 =d; ¡ s ¡ 2 =d; s ¡ 3 =d; : : : ; s ( d ¡ 2) =d; ¡ s ¡ ( d ¡ 1) =d ) : (3. 43) If r = d ¡ 1 and d is odd, then ther e is only one idempotent and it is given by (3. 42). If r = 1 and d is e ven, ther e ar e two possible idempotents: e = toep (1 =d; s ¡ 1 =d; s ¡ 2 =d; : : : ; s ¡ d ¡ 2 =d; s ¡ ( d ¡ 1) =d ) (3. 44) or e = toep (1 =d; ¡ s ¡ 1 =d; s ¡ 2 =d; : : : ; s ¡ ( d ¡ 2) =d; ¡ s ¡ ( d ¡ 1) =d ) : (3. 45) If r = 1 and d is odd, then ther e is only one idempotent and it is given by (3. 44). Using the spectral representation (3. 40), we observ e that Lemma 3.4 still holds for a compact abelian group in S d ( t ) : F ollo wing the same procedures as in earlier sections, we ag ain will ha v e analogous results for S d ( t ) : W e omit the details. 78

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Chapter 4 Numerical Calculation of L yapuno v Exponents for Some Random Fibonacci Recurrences 4.1 Random Matrix Products and Random Fibonacci Sequences W e introduce here some concepts from the theory of random matrix products that will be useful in the sequel, specializing in the case of matrices in M = M (2 ; R ) ; the space of 2 £ 2 real matrices. F or a (column) v ector x = 0@ a b 1A 2 R 2 and Y 2 M we let k x k = p a 2 + b 2 ; k Y k = sup fk Y x k : x 2 R 2 ; k x k = 1 g : Let A 1 ; A 2 ; : : : be a sequence of i.i.d. random matrices in M with common distrib ution such that the e xpectation E (log + k A 1 k ) ; where f + := sup( f ; 0) ; is nite. It is then clear from the subadditi vity of the sequence f E (log k A n A n ¡ 1 ¢ ¢ ¢ A 1 k ) : n 1 g ; that the limit lim n !1 E (log k A n A n ¡ 1 ¢ ¢ ¢ A 1 k ) n e xists in R [ f¡1g : This limit is called the (upper) L yapuno v e xponent associated with (or with the random sequence f A n : n 1 g ), and is denoted by : A well-kno wn result by Furstenber g and K esten [FK] gi v es the stronger result = lim n !1 1 n log k A n A n ¡ 1 ¢ ¢ ¢ A 1 k (4. 1) almost surely 79

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As an immediate application of Furstenber g and K esten' s result, consider the random sequence 8<: x n = 0@ a n b n 1A : n 1 9=; of column v ectors in R 2 such that x n = A n x n ¡ 1 ; n 1 ; with x 0 = 0@ a 0 b 0 1A x ed, and A 1 ; A 2 ; : : : is a sequence of i.i.d. random matrices in M with associated L yapuno v e xponent : Then x n = A n A n ¡ 1 ¢ ¢ ¢ A 1 x 0 ; and therefore, almost surely lim sup n !1 j a n j 1 =n lim sup n !1 k x n k 1 =n lim sup n !1 k A n A n ¡ 1 ¢ ¢ ¢ A 1 k 1 =n q a 20 + b 20 1 =n = e ; so that lim sup n !1 log j a n j n (4. 2) almost surely The inequality lim inf n !1 log j a n j n (4. 3) holds almost surely as well, although it is not straightforw ard to pro v e (see [BL]). Combining (4. 2) and (4. 3) then yields = lim n !1 log j a n j n ; (4. 4) almost surely Ne xt, we introduce Furstenber g' s Theorem, one of whose conclusions is a nice inte gral formula for the L yapuno v e xponent associated with certain random matrix sequences. T w o nonzero v ectors x; y 2 R 2 ha v e the same direction if for some 2 R ; x = y : This denes an equi v alence relation on R 2 n f 0 g ; and each equi v alence class under is called 80

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a direction The set of directions is called the projecti v e space denoted by P = P ( R 2 ) : F or x 2 R 2 n f 0 g ; let x 2 P denote its direction. Let G = GL (2 ; R ) be the group of nonsingular matrices in M Observ e that G acts on P via Y ¢ x = Y x; where Y 2 G and and x 2 R 2 n f 0 g : This allo ws us to dene a “con v olution” product ¤ between a probability measure on G and a probability measure on P ; dened as the distrib ution on P satisfying Z P f ( x ) d ( ¤ )( x ) = Z P Z G f ( Y ¢ x ) d ( Y ) d ( x ) for an y bounded Borel function f on P : is said to be -in v ariant if ¤ = : is said to be continuous if ( f x g ) = 0 for all x 2 P ( R 2 ) : W e no w state without proof Furstenber g' s Theorem (see, for e xample, [BL], pp. 53-54): T H E O R E M 4.1 Let be a pr obability measur e on G; the gr oup of 2 £ 2 in vertible r eal matrices. Let G be the smallest closed subgr oup of G whic h contains the support of : Suppose that the following hold: (i) for Y 2 G ; j det Y j = 1 ; (ii) G is not compact, and (iii) for any x 2 R 2 n f 0 g ; jf Y x : Y 2 G gj 2 : Then if f Y 1 ; Y 2 ; : : : g is a sequence of independent -distrib uted r andom matrices in G with E (log + k Y 1 k ) < 1 and as its associated L yapuno v e xponent, = lim n !1 1 n log k Y n ¢ ¢ ¢ Y 1 x k > 0 for any x 2 R 2 n f 0 g ; Mor eo ver ther e e xists a unique continuous -in variant measur e on P = P ( R 2 ) suc h that = Z P Z G log k Y x k k x k d ( Y ) d ( x ) : (4. 5) 81

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No w we recall the three types of random Fibonacci sequences we introduced in Chapter 1: x n +1 = x n § x n ¡ 1 ; (4. 6) x n +1 = § x n + x n ¡ 1 ; (4. 7) and x n +1 = x n § x n ¡ 1 ; (4. 8) where x 0 = x 1 = 1 ; each § sign is chosen independently and + and ¡ occur with probabilities p and q := 1 ¡ p; respecti v ely with 0 < p < 1 : In (4. 8), > 0 is a x ed “gro wth parameter ” W e refer to (4. 6) and (4. 7) as “V isw anath-type” and to (4. 8) as “Embree-T refethen-type” random Fibonacci sequences. F or a x ed v alue of p; let us denote the L yapuno v e xponents of the random sequences (4. 6), (4. 7), and (4. 8) by 1 ; 2 ; and ; respecti v ely It is immediate from Furstenber g and K esten' s result, as we shall see belo w that, in f act, 1 ; 2 ; and are equal to the almost sure limit lim n !1 log j x n j n with x n gi v en by (4. 6), (4. 7), and (4. 8), respecti v ely Our aim in this chapter is to numerically in v estig ate the beha vior of 1 ; 2 ; and as a function of p: W e follo w [V i ] for the theoretical part and [ET] for the numerical part of our solution. W e be gin with the V isw anath-type recurrences (4. 6) and (4. 7) and e xpress these equations using matrices. F or (4. 6) we ha v e the matrix equation 0@ x n x n +1 1A = 0@ 0 1 § 1 1 1A 0@ x n ¡ 1 x n 1A = M (1) n M (1) n ¡ 1 ¢ ¢ ¢ M (1) 1 0@ 11 1A ; (4. 9) where M (1) 1 ; M (1) 2 ; : : : in (4. 9) are i.i.d. matrices such that M (1) 1 is either A + = 0@ 0 1 1 1 1A 82

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with probability p; or A ¡ = 0@ 0 1 ¡ 1 1 1A with probability q : F or (4. 7), the corresponding matrix formulation is gi v en by 0@ x n x n +1 1A = 0@ 0 1 1 § 1 1A 0@ x n ¡ 1 x n 1A = M (2) n M (2) n ¡ 1 ¢ ¢ ¢ M (2) 1 0@ 11 1A ; (4. 10) where M (2) 1 ; M (2) 2 ; : : : in (4. 10) are i.i.d. matrices such that M (2) 1 is either B + = A + with probability p; or B ¡ = 0@ 0 1 1 ¡ 1 1A with probability q : Since the matrix products appearing in (4. 9) and (4. 10) are products of i.i.d. random matrices, it follo ws from Furstenber g and K esten' s result that 1 (resp. 2 ) is, in f act, equal to the (upper) L yapuno v e xponent associated with the random sequence M (1) 1 ; M (1) 2 ; : : : (resp. M (2) 1 ; M (2) 2 ; : : : ). [BL]. And since Furstenber g' s theorem applies in this case, we obtain, from equation (4. 5), the inte gral formulas 1 = Z 1 ¡1 a ( m; p; 1) d 1 ( m ) ; (4. 11) and 2 = Z 1 ¡1 a ( m; p; 1) d 2 ( m ) ; (4. 12) where a ( m; p; t ) = 1 2 p log m 2 + ( m + t ) 2 m 2 + ( m ¡ t ) 2 + log m 2 + ( m ¡ t ) 2 1 + m 2 ; (4. 13) and where 1 and 2 are the unique in v ariant continuous probability measure for the random w alk on directions x in the plane (parametrized either by slopes m 2 ( ¡1 ; 1 ] or by angles 83

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2 ( ¡ = 2 ; = 2] ) induced respecti v ely by the probability distrib utions 1 and 2 on 2 £ 2 (real) matrices satisfying 1 ( A + ) = p; 1 ( A ¡ ) = q ; (4. 14) and 2 ( B + ) = p; 2 ( B ¡ ) = q : (4. 15) The problem, ho we v er is that we do not ha v e a closed form e xpression for the in v ariant measures 1 and 2 to e v aluate the inte grals (4. 11) and (4. 12) when p 6 = 1 = 2 : Thus, we proceed numerically as follo ws. Since 1 and 2 satisfy 1 = 1 ¤ 1 and 2 = 2 ¤ 2 ; and both are measures dened on Borel subsets of R (as slopes) or ( ¡ = 2 ; = 2] (as angles), we ha v e the in v ariance equations 1 ([ a; b ]) = p 1 1 [ a; b ] ¡ 1 + q 1 1 ¡ [ a; b ] + 1 ; (4. 16) and 2 ([ a; b ]) = p 2 1 [ a; b ] ¡ 1 + q 2 1 [ a; b ] + 1 ; (4. 17) for an y slope interv al [ a; b ] with § 1 62 ( a; b ) ; or the corresponding equations 1 ( I ) = p 1 tan ¡ 1 1 (tan I ) ¡ 1 + q 1 tan ¡ 1 1 ¡ (tan I ) + 1 ; (4. 18) and 2 ( I ) = p 2 tan ¡ 1 1 (tan I ) ¡ 1 + q 2 tan ¡ 1 1 (tan I ) + 1 ; (4. 19) for an y angular interv al I ( ¡ = 2 ; = 2] with § 4 62 I : The b ulk of the numerical calculation is done by “discretizing” equations (4. 18) and (4. 19). W e describe the details of this process in the ne xt section. 84

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The same theoretical calculations may be done for random recurrences (4. 8) of EmbreeT refethen type. W e skip the details and just present the rele v ant analogous quantities. W ith x n as in (4. 8), and the same assumption of independence of choice of + and ¡ signs with the same respecti v e probabilities p and q as before, and for x ed > 0 ; we obtain as the in v ariant measure satisfying ( I ) = p tan ¡ 1 (tan I ) ¡ 1 + q tan ¡ 1 ¡ (tan I ) + 1 (4. 20) for an y angular interv al I ( ¡ = 2 ; = 2] with § 4 62 I : As before, we use Furstenber g' s Theorem to arri v e at the inte gral formula = Z 1 ¡1 a ( m; p; ) d ( m ) ; (4. 21) with a ( m; p; ) as in (4. 13). In the ne xt section we describe the numerical approximation of 1 ; 2 ; and for > 0 : 4.2 Numerical Calculation of the L yapuno v Exponents The rst step in the numerical calculation of the L yapuno v e xponents 1 ; 2 ; and ; is the numerical approximation of the corresponding in v ariant measures 1 ; 2 ; and : W e follo w the ideas in [ET ] for this numerical approximation. W e subdi vide the interv al [ ¡ = 2 ; = 2] into N = 2 n equally spaced angular interv als I 1 ; I 2 ; : : : ; I N ; each of length ¢ = = N : W e then approximate each of the angular inv ariance equations (4. 18), (4. 19), and (4. 20) on the discrete set of interv als I j = [ ¡ 2 + ( j ¡ 1)¢ ; ¡ 2 + j ¢] ; j = 1 ; : : : ; N ; as follo ws. Let g be the map g ( x; ) = tan ¡ 1 tan x ¡ 1 : Observ e g ( x; ) is continuous for all x 2 ( ¡ = 2 ; = 2] e xcept at x = = 4 : W e can then write (4. 18), (4. 19), and (4. 20) as 1 ( I ) = p 1 ( g ( I ; 1)) + q 1 ( ¡ g ( I ; 1)) ; (4. 22) 85

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2 ( I ) = p 2 ( g ( I ; 1)) + q 2 ( ¡ g ( ¡ I ; 1)) (4. 23) and ( I ) = p ( g ( I ; )) + q ( ¡ g ( I ; )) ; (4. 24) respecti v ely Let ` ( I ) denote the length of the angular interv al I ( ¡ = 2 ; = 2] : Then, writing ( j ) 1 ; ( j ) 2 ; and ( j ) to denote the discrete approximations to 1 ( I j ) ; 2 ( I j ) ; and ( I j ) ; respecti v ely we ha v e the equations ( j ) 1 = 1 ¢ N X k =1 p ` ¡ I k \ g ( I j ; 1) ¢ + q ` ¡ I k \ ¡ g ( I j ; 1) ¢ ( k ) 1 ; (4. 25) ( j ) 2 = 1 ¢ N X k =1 p ` ¡ I k \ g ( I j ; 1) ¢ + q ` ¡ I k \ ¡ g ( ¡ I j ; 1) ¢ ( k ) 2 ; (4. 26) and ( j ) = 1 ¢ N X k =1 p ` ¡ I k \ g ( I j ; ) ¢ + q ` ¡ I k \ ¡ g ( I j ; ) ¢ ( k ) (4. 27) for j = 1 ; 2 ; : : : ; N : In the system (4. 25) of N linear equations in the N unkno wns ( k ) 1 ; k = 1 ; 2 ; : : : ; N ; the quantity in parentheses found on the right side of the j th equation, j = 1 ; 2 ; : : : ; N ; represents the amount of o v erlap between I k and g ( I j ; 1) and between I k and ¡ g ( I j ; 1) : The corresponding quantities in systems (4. 26) and (4. 27) are e xplained similarly All three N £ N linear systems (4. 25), (4. 26), and (4. 27) are of rank N ¡ 1 ; which can be made consistent by replacing the N th equation by the respecti v e conserv ation conditions N X j =1 ( j ) 1 = 1 ; N X j =1 ( j ) 2 = 1 ; N X j =1 ( j ) = 1 : (4. 28) W e also see that these linear systems are sparse: the length ` ( g ( I j ; )) of the image of the j th subinterv al I j is at most ¢ times the maximum v alue of j g 0 ( x; ) j for x 2 [ ¡ = 2 ; = 2] ; which is 2 + 2 + p 4 + 4 2 : (4. 29) 86

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Therefore, the number of nonzero coef cients in (4. 27) is O ( ) if > 1 ; and O (1 = ) if < 1 : If = 1 ; then (4. 29) e v aluates to 1 + g ; where g = 1+ p 5 2 is the golden ratio, and consequently there are at most 4 nonzero coef cients in (4. 25) or (4. 26). Thus, in the end, our numerical approximation to the in v ariant measures 1 ; 2 ; and consists of solving the sparse linear systems (4. 25) to (4. 27) together with their respecti v e conserv ation conditions displayed in (4. 28). W e solv ed these linear systems using Mathematica 5.2' s b uilt-in sparse systems solv er taking N to 2 21 when p = 1 = 2 to compare our results with the numerical v alues reported by Embree and T refethen in [ET ]. F or this part of the computational process, we w ould lik e to ackno wledge the use of the services pro vided by Research Computing, Uni v ersity of South Florida. Once we found that our results were consistent with Embree and T refethen' s, we lo wered the v alue of N to 256 for all other v alues of p in the calculations that produced the gures we report here. Figures 1 and 2 sho w the respecti v e graphs – actually histograms – of the in v ariant measures 1 and 2 computed in the manner just described for p = 0 : 1 ; 0 : 2 ; : : : ; 0 : 9 : Figures 3 to 6 display the same information for in v ariant measures for = 1 = 2 ; 3 = 4 ; 2 ; and 8 ; respecti v ely These histograms are based on N = 256 equally sized subdi visions of [ ¡ = 2 ; = 2] : The v ertical axis in each graph represents the approximated v alue of the in v ariant measures on the subinterv als I k ; k = 1 ; 2 ; : : : ; N : Se v eral interesting observ ations may be deri v ed from these graphs. First, it is e vident from Figure 1 that 1 e xhibits a reection property with respect to p = 1 = 2 : 1 ( I k ; p ) = 1 ( ¡ I k ; 1 ¡ p ) : (4. 30) Also, Figure 2 suggests that the smoothness of the in v ariant measure 2 appears to decrease as p increases. This beha viour is similarly observ ed for for a x ed and, moreo v er does not seem to depend on : On the other hand, the opposite situation seems to occur when p is x ed and is allo wed 87

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to v ary Figures 7 to 10 sho w ho w beha v es for dif ferent v alues of < 1 when p is x ed at 0 : 2 ; 0 : 4 ; 0 : 6 ; and 0 : 8 ; respecti v ely Figures 11 to 14 display the same information, b ut for v alues of > 1 : 1.5 1 0.5 0 0.5 1 1.5 p = 0.7 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.8 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.9 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.4 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.5 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.6 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.1 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.2 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 p = 0.3 0.5 1 1.5 2 2.5 3 Figure 1: 1 ( p ) for dif ferent v alues of p 88

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1.5 1 0.5 0 0.5 1 1.5 p = 0.7 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.8 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.9 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.4 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.5 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.6 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.1 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.2 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.3 0.5 1 1.5 2 2.5 3 3.5 4 Figure 2: 2 ( p ) for dif ferent v alues of p 89

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1.5 1 0.5 0 0.5 1 1.5 p = 0.7 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.8 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.9 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.4 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.5 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.6 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.1 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.2 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 p = 0.3 1 2 3 4 5 6 7 Figure 3: ( p ) ; where = 1 = 2 ; for dif ferent v alues of p 90

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1.5 1 0.5 0 0.5 1 1.5 p = 0.7 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.8 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.9 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.4 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.5 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.6 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.1 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.2 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 p = 0.3 0.5 1 1.5 2 2.5 3 3.5 4 Figure 4: ( p ) ; where = 3 = 4 ; for dif ferent v alues of p 91

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1.5 1 0.5 0 0.5 1 1.5 p = 0.7 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.8 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.9 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.4 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.5 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.6 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.1 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.2 0.25 0.5 0.75 1 1.25 1.5 1.5 1 0.5 0 0.5 1 1.5 p = 0.3 0.25 0.5 0.75 1 1.25 1.5 Figure 5: ( p ) ; where = 2 ; for dif ferent v alues of p 92

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1.5 1 0.5 0 0.5 1 1.5 p = 0.7 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.9 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 1 0.5 0 0.5 1 1.5 p = 0.3 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 6: ( p ) ; where = 8 ; for dif ferent v alues of p 93

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1.5 1 0.5 0 0.5 1 1.5 b = 0.7 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 b = 0.8 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 b = 0.9 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 b = 0.4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 0.5 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 0.6 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 0.1 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 0.2 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 0.3 1 2 3 4 5 Figure 7: ( p ) for x ed p = 0 : 2 and dif ferent v alues of < 1 94

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1.5 1 0.5 0 0.5 1 1.5 b = 0.7 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 0.8 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 0.9 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 0.4 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 b = 0.5 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 b = 0.6 0.5 1 1.5 2 2.5 3 3.5 4 1.5 1 0.5 0 0.5 1 1.5 b = 0.1 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 b = 0.2 1 2 3 4 5 6 7 1.5 1 0.5 0 0.5 1 1.5 b = 0.3 1 2 3 4 5 6 7 Figure 8: ( p ) for x ed p = 0 : 4 and dif ferent v alues of < 1 95

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1.5 1 0.5 0 0.5 1 1.5 b = 0.7 1 2 3 4 5 6 7 8 1.5 1 0.5 0 0.5 1 1.5 b = 0.8 1 2 3 4 5 6 7 8 1.5 1 0.5 0 0.5 1 1.5 b = 0.9 1 2 3 4 5 6 7 8 1.5 1 0.5 0 0.5 1 1.5 b = 0.4 2 4 6 8 10 12 1.5 1 0.5 0 0.5 1 1.5 b = 0.5 2 4 6 8 10 12 1.5 1 0.5 0 0.5 1 1.5 b = 0.6 2 4 6 8 10 12 1.5 1 0.5 0 0.5 1 1.5 b = 0.1 2 4 6 8 10 12 14 16 1.5 1 0.5 0 0.5 1 1.5 b = 0.2 2 4 6 8 10 12 14 16 1.5 1 0.5 0 0.5 1 1.5 b = 0.3 2 4 6 8 10 12 14 16 Figure 9: ( p ) for x ed p = 0 : 6 and dif ferent v alues of < 1 96

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1.5 1 0.5 0 0.5 1 1.5 b = 0.7 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 0.8 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 0.9 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 0.4 2 4 6 8 10 1.5 1 0.5 0 0.5 1 1.5 b = 0.5 2 4 6 8 10 1.5 1 0.5 0 0.5 1 1.5 b = 0.6 2 4 6 8 10 1.5 1 0.5 0 0.5 1 1.5 b = 0.1 2.5 5 7.5 10 12.5 15 17.5 20 1.5 1 0.5 0 0.5 1 1.5 b = 0.2 2.5 5 7.5 10 12.5 15 17.5 20 1.5 1 0.5 0 0.5 1 1.5 b = 0.3 2.5 5 7.5 10 12.5 15 17.5 20 Figure 10: ( p ) for x ed p = 0 : 8 and dif ferent v alues of < 1 97

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1.5 1 0.5 0 0.5 1 1.5 b = 64 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 128 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 256 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 8 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 16 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 32 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 1 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 b = 2 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 b = 4 0.2 0.4 0.6 0.8 1 Figure 11: ( p ) for x ed p = 0 : 2 and dif ferent v alues of > 1 98

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1.5 1 0.5 0 0.5 1 1.5 b = 64 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 128 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 256 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 8 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 16 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 32 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 1 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 b = 2 0.5 1 1.5 2 2.5 3 1.5 1 0.5 0 0.5 1 1.5 b = 4 0.5 1 1.5 2 2.5 3 Figure 12: ( p ) for x ed p = 0 : 4 and dif ferent v alues of > 1 99

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1.5 1 0.5 0 0.5 1 1.5 b = 64 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 128 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 256 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 8 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 16 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 32 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.5 1 0.5 0 0.5 1 1.5 b = 1 1 2 3 4 5 6 1.5 1 0.5 0 0.5 1 1.5 b = 2 1 2 3 4 5 6 1.5 1 0.5 0 0.5 1 1.5 b = 4 1 2 3 4 5 6 Figure 13: ( p ) for x ed p = 0 : 6 and dif ferent v alues of > 1 100

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1.5 1 0.5 0 0.5 1 1.5 b = 64 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 128 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 256 1 2 3 4 5 1.5 1 0.5 0 0.5 1 1.5 b = 8 1 2 3 4 5 6 1.5 1 0.5 0 0.5 1 1.5 b = 16 1 2 3 4 5 6 1.5 1 0.5 0 0.5 1 1.5 b = 32 1 2 3 4 5 6 1.5 1 0.5 0 0.5 1 1.5 b = 1 2 4 6 8 10 12 14 16 1.5 1 0.5 0 0.5 1 1.5 b = 2 2 4 6 8 10 12 14 16 1.5 1 0.5 0 0.5 1 1.5 b = 4 2 4 6 8 10 12 14 16 Figure 14: ( p ) for x ed p = 0 : 8 and dif ferent v alues of > 1 101

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Finally once we ha v e the approximation to the in v ariant measures 1 ; 2 ; and ; the cor responding L yapuno v e xponents 1 ; 2 ; and may be calculated by numerical inte gration applied to (4. 11), (4. 12), and (4. 21), respecti v ely Figures 15 and 16 sho w the L yapuno v e xponents 1 and 2 vs. p for 200 v alues of p between 0 and 1 : Note ho w 2 increases as p increases, whereas 1 e xhibits symmetry with respect to p = 1 = 2 : 0 0.2 0.4 0.6 0.8 1 p 0.1 0.2 0.3 0.4 0.5 g2 Figure 15: 1 vs. p 0 0.2 0.4 0.6 0.8 1 p 0.1 0.2 0.3 0.4 0.5 g2 Figure 16: 2 vs. p 102

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Figure 17 sho ws for = 0 : 5 ; 0 : 75 ; 2 ; and 8 : In the gure, dark er curv es correspond to smaller v alues of : It appears that for v alues of 1 ; > 0 (hence the corresponding random recurrence (4. 8) gro ws e xponentially) no matter what p is. On the other hand, for v alues of < 1 ; the phenomenon observ ed by Embree and T refethen in [ET ] appears to ha v e an analogue: there e xists some p ¤ = p ¤ ( ) for which ( p ¤ ) = 0 ; which means the corresponding random recurrence (4. 8) neither gro ws nor decays. 0 0.2 0.4 0.6 0.8 1 p -0.25 0 0.25 0.5 0.75 1 gb Figure 17: vs. p for = 0 : 5 ; 0 : 75 ; 2 ; and 8 103

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References [BL] P Bougerol and J. Lacroix, Random Products of Matrices with Applications to Innite-Dimensional Schr ¨ odinger Operators, Birkh ¨ auser Basel, 1984. [CR] S. Chakraborty and B. V Rao, “Con v olution po wers of probabilities on stochastic matrices of order 3, ” Sankh y a: The Indian Journal of Statistics, V ol. 60, Series A, Pt. 2, 151–170, 1998. [D] P J. Da vis, Cir culant Matrices John W ile y & Sons, Ne w Y ork, 1979. [Di] P Diaconis, “P atterned Matrices, ” Proceedings of Symposia in Applied Mathematics 40, 37–58, 1990. [DM] S. Dhar and A. Mukherjea, “Con v er gence in Distrib ution of Products of I.I.D. Nonne g ati v e Matrices, ” Journal of Theoretical Probability V ol. 10, No. 2, 375–393, 1997. [ET] M. Embree and L. T refethen, “Gro wth and Decay of Random Fibonacci Sequences, ” Proc. Ro y Soc. London Ser A, Math. Ph ys. Eng. Sci., V olume 455, 2471-2485, 1999. [FK] H. Furstenber g and H. K esten, “Products of Random Matrices, ” Ann. Math. Stat. 31, 457–469, 1960. [HMu] G. H ¨ ogn ¨ as and A. Mukherjea, Pr obability Measur es on Semigr oups Plenum Press, Ne w Y ork, 1995. [HMi] F Y Hunt, and W M. Miller “On the approximation of in v ariant measures, ” J. Stat. Ph ys., V ol. 66, No. 1/2, 535-548, 1992. 104

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[LM] C. C. Lo and A. Mukherjea, “Con v er gence in Distrib ution of Products of d by d Random Matrices, ” Journal of Mathematical Analysis and Applications, V ol. 162, No. 1, 71–91, 1991. [M1] A. Mukherjea, “Limit theorems: stochastic matrices, er godic mark o v chains and measures on semigroups, ” Probabilistic Analysis and Related T opics, V ol. 2, 143-203, 1979. [M2] A. Mukherjea, “Completely Simple Semigroups of Matrices, ” Semigroup F orum, V ol. 33, 405-429, 1986. [Pe] Y Peres, “ Analytic dependence of L yapuno v e xponents on transition probabilities, ” L yapuno v Exponents, Lecture Notes in Math. 1486, Springer -V erlag, Berlin, 1986. [MR1] M. Rosenblatt, Mark o v Pr ocesses: Structur e and Asymptotic Behavior Springer V erlag, 1971. [MR2] M. Rosenblatt, “Products of i.i.d. stochastic matrices, ” Journal of Mathematical Analysis and Applications, V ol. 11, 1–10, 1965. [V i] D. V isw anath, “Random Fibonacci Sequences and the Number 1 : 13198824 ::: ” Mathematics of Computation, V olume 69, Number 231, 1131–1155, 2000. 105

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About the Author Edg ardo Cure g is a nati v e of the Philippines. In 1984 he entered the Uni v ersity of the Philippines, initially majoring in Computer Science. In 1986, after taking a course in Abstract Algebra under the illustrious Dr Aurora T rance, he changed his major to Mathematics. He went on to study Information Engineering in Japan from 1988 to 1996 under scholarships of fered by Japan' s Monb usho (no w called the Ministry of Education, Culture, Sports, Science and T echnology or MEXT) and IBM Asia. He went back to the Philippines in 1997 and taught Mathematics at De La Salle Uni v ersity in Manila. In 2000, he relocated to the United States and be g an his graduate education in Mathematics at the Uni v ersity of South Florida, where, in August 2001, he w as a w arded a Master of Arts de gree. That same year he w as a w arded a three-year Graduate Research Fello wship by the National Science F oundation. Since his admission to doctoral candidac y in 2003 he has been conducting research on products of random matrices under the supervision of Dr Aruna v a Mukherjea.