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

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 [1], [2] and [3]. 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.