A Transient Random Walk on Stochastic Matrices with Dirichlet Distributions

Abstract

Let $X_1$ be a $(d \times d)$ random stochastic matrix such that the rows of $X_1$ are independent, with Dirichlet distributions. The rows of the $(d \times d)$ matrix $A$ are the parameters of these Dirichlet distributions, and we assume that the sums of the rows and columns of $A$ provide the same vector $r = (r_1,\ldots,r_d)$. If $(X_n)^\infty_{n=1}$ are i.i.d., we prove that $\lim_{n\rightarrow\infty}(X_n \cdots X_1)$ almost surely has identical rows, which are Dirichlet distributed with parameter $r$. Van Assche has proved this for $d = 2$ and four identical entries for $A$.