  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.
