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.
Santanu Chakraborty^{1} Arunava Mukherjea^{2} ^{}
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November 2019
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