• Arunava Mukherjea

      Articles written in Proceedings – Mathematical Sciences

    • Upper Packing Dimension of a Measure and the Limit Distribution of Products of i.i.d. Stochastic Matrices

      Arunava Mukherjea Ricardo Restrepo

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      This article gives sufficient conditions for the limit distribution of products of i.i.d. $2\times 2$ stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is countably infinite. It extends a previous result for which the support of the random matrices is finite. The result is based on adapting existing proofs in the context of attractors and iterated function systems to the case of infinite iterated function systems.

    • Limit distributions of random walks on stochastic matrices

      Santanu Chakraborty Arunava Mukherjea

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      Problems similar to Ann. Prob. 22 (1994) 424–430 and J. Appl. Prob. 23 (1986) 1019–1024 are considered here. The limit distribution of the sequence $X_{n}X_{n−1}\ldots X_{1}$, where $(X_{n})_{n\geq 1}$ is a sequence of i.i.d. $2 \times 2$ stochastic matrices with each $X_{n}$ distributed as 𝜇, is identified here in a number of discrete situations. A general method is presented and it covers the cases when the random components $C_{n}$ and $D_{n}$ (not necessarily independent), $(C_{n}, D_{n})$ being the first column of $X_{n}$, have the same (or different) Bernoulli distributions. Thus $(C_{n}, D_{n})$ is valued in $\{0, r\}^{2}$, where 𝑟 is a positive real number. If for a given positive real 𝑟, with $0 \lt r \leq \frac{1}{2}$, $r^{-1}C_{n}$ and $r^{-1}D_{n}$ are each Bernoulli with parameters $p_{1}$ and $p_{2}$ respectively, $0 < p_{1}$, $p_{2} \lt 1$ (which means $C_{n}\sim p_{1}\delta_{\{r\}} + (1 - p_{1})\delta_{\{0\}}$ and $D_{n} \sim p_{2}\delta_{\{r\}} + (1 - p_{2})\delta_{\{0\}}$), then it is well known that the weak limit 𝜆 of the sequence $\mu^{n}$ exists whose support is contained in the set of all $2 \times 2$ rank one stochastic matrices. We show that $S(\lambda)$, the support of 𝜆, consists of the end points of a countable number of disjoint open intervals and we have calculated the 𝜆-measure of each such point. To the best of our knowledge, these results are new.

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