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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 isDirichlet distributed. This extends a result by Chamayou and Letac (1994) andis illustrated by several examples that are of interest in applications.

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