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

TL;DR

This paper proves convergence conditions for infinite products of stochastic matrices with positive diagonals, relevant to opinion dynamics and Markov processes, with some improvements and related results discussed.

Contribution

It establishes new convergence criteria for infinite products of stochastic matrices with positive diagonals, extending previous results and connecting to opinion dynamics models.

Findings

Convergence occurs if minimal positive entries do not decay too fast.

Zero-entry symmetry or bounded subproduct length ensures convergence.

Provides small improvements and links to inhomogeneous Markov processes.

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.