• 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

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

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, $0 &lt; p_{1}$, $p_{2} \lt 1$ (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.

• The convolution equation $\sigma*\mu=\mu$ on non-compact non-abelian semigroups

In probability theory, often in connection with problems on weak convergence, and also in other contexts, convolution equations of the form $\sigma*\mu=\mu$ come up. Many years ago, Choqet and Deny (C. R. Acad. Sci. Paris 250 (1960) 799-801) studied these equations in locally compact abelian groups. Later, Szekely and Zeng (J. Theoret. Probab. 3(2) (1990) 361-365) studied these equations in abelian semigroups. Like in [2], the results in [7] are also complete. Thus, these equations are studied here for the first time on non-compact non-abelian semigroups. Our main results are Theorems 3.1 and 3.3 in section 3. They are new results as far as we know, and also the best possible under a minor condition. All semigroups in this paper are, unless otherwise mentioned, locally compact Hausdorff second countable topological semigroups. Theorems 3.1 and 3.3 hold for these semigroups.

• # Proceedings – Mathematical Sciences

Volume 130, 2020
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• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019