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

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 independent and identically distributed copies X(k) of X converge almost surely as n → ∞ 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.