Convergence in distribution of products of i. i. d. nonnegative matrices (Q1368994)

A rather old problem of weak convergence of stochastic matrices is discussed. Let \(\mu^n\) stand for \(n\)th convolution of a distribution \(\mu\) defined on the set of \(m\times m\) stochastic matrices. In the case \(m=2\) it is well-known that the sequence \(\{\mu^n\}_{n\geq 1}\) weakly converges if and only if \(\mu\) is not the unit mass at the stochastic matrix where the diagonal elements are all zeros. In the case \(m=3\), the authors give necessary and sufficient conditions in terms of the support on \(\mu\) for weak convergence of the sequence \(\{\mu^n\}_{n\geq 1}\). They also point out a very general sufficient condition in terms of support of \(\mu\) for each \(m\geq 3\).