On Backward Product of Stochastic Matrices

TL;DR

This paper investigates the ergodic behavior of backward products of stochastic matrices, introducing the absolute infinite flow property, and establishes conditions for convergence and stability in such matrix chains.

Contribution

It introduces the absolute infinite flow property and proves its equivalence to ergodicity for doubly stochastic chains, along with convergence rate and stability conditions.

Findings

Absolute infinite flow property is necessary for ergodicity.

The property is equivalent to ergodicity in doubly stochastic chains.

Product of doubly stochastic matrices converges up to permutation.

Abstract

We study the ergodicity of backward product of stochastic and doubly stochastic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of convergence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochastic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.