# Some problems on products of random matrices

2006

Dissertation

Ph.D.

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Arunava Mukherjea, Ph.D.

## Keywords

Weak convergence, Stochastic matrices, Circulant matrices, Lyapunov exponents, Random Fibonacci sequences

## 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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