Convolution powers of probabilities on stochastic matrices (Q5939317)

Let \(S_d\) denote a semigroup of \(d\times d\) stochastic matrices, \(d\in\mathbb{N}\). Let \(\mu\) be a probability on \(S_d\) and \(\mu^n\) denote the \(n\)-fold convolution of \(\mu\). The aim of the paper is to derive a necessary and sufficient condition for the convergence of the sequence \(\mu^n\). More precisely, let \(G\) be the closed semigroup generated by the support of \(\mu\), and let \(\rho\) be an associated probability induced by \(\mu\) on a finite subgroup \(H\) that is related to the kernel of \(G\). \(H\) is actually a subgroup of the permutation group obtained as a decomposition of the kernel of \(G\). Now, the main result proves that the sequence \(\mu^n\) converges on \(S_d\) iff \(\rho^n\) converges on the (sub)group \(H\).