A characterisation of transient random walks on stochastic matrices with Dirichlet distributed limits

TL;DR

This paper characterizes the distributions of random stochastic matrices whose infinite products converge almost surely to a Dirichlet distribution, extending previous results and providing practical examples.

Contribution

It extends the characterization of stochastic matrices with Dirichlet limits, broadening understanding of their convergence properties and applications.

Findings

Identifies conditions for convergence to Dirichlet distribution

Provides examples of such stochastic matrices

Extends previous theoretical results

Abstract

We characterise the class of distributions of random stochastic matrices $X$ with the property that the products $X(n)X(n-1) ... X(1)$ of i.i.d. copies $X(k)$ of $X$ converge a.s. as $n \rightarrow \infty$ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.