Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics background

Abstract

We present a convergence result for infinite products of stochastic matrices with positive diagonals. We regard infinity of the product to the left. Such a product converges partly to a fixed matrix if the minimal positive entry of each matrix does not converge too fast to zero and if either zero-entries are symmetric in each matrix or the length of subproducts which reach the maximal achievable connectivity is bounded. Variations of this result have been achieved independently in Lorenz 2005, Moreau 2005 and Hendrickx 2005. We present briefly the opinion dynamics context, discuss the relations to infinite products where infinity is to the right (inhomogeneous Markov processes) and present a small improvement and sketch another.