On Approximations and Ergodicity Classes in Random Chains

TL;DR

This paper investigates the limiting behavior of non-ergodic random chains by introducing l_1-approximations and infinite flow graphs, linking graph connectivity to ergodic class structure, with applications to dynamic networks.

Contribution

It introduces a novel approach using l_1-approximations and infinite flow graphs to identify ergodic classes in random chains, extending understanding of their limiting behavior.

Findings

l_1-approximations preserve certain limiting behaviors

Connectivity of the infinite flow graph relates to ergodic group structure

Conditions are provided to identify ergodicity groups via graph components

Abstract

We study the limiting behavior of a random dynamic system driven by a stochastic chain. Our main interest is in the chains that are not necessarily ergodic but rather decomposable into ergodic classes. To investigate the conditions under which the ergodic classes of a model can be identified, we introduce and study an l_1-approximation and infinite flow graph of the model. We show that the l_1-approximations of random chains preserve certain limiting behavior. Using the l_1-approximations, we show how the connectivity of the infinite flow graph is related to the structure of the ergodic groups of the model. Our main result of this paper provides conditions under which the ergodicity groups of the model can be identified by considering the connected components in the infinite flow graph. We provide two applications of our main result to random networks, namely broadcast over time-varying networks and networks with random link failure.