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

We characterise the class of distributions of random stochastic matrices $X$ with the property that the products $X(n)X(n-1)\mathrm{.}\mathrm{.}\mathrm{.}X(1)$ of i.i.d. copies $X(k)$ of $X$ converge a.s. as $n\to \mathrm{\infty }$ 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.