Merging and stability for time inhomogeneous finite Markov chains

TL;DR

This paper explores the concepts of merging and stability in time inhomogeneous finite Markov chains, analyzing how these chains forget initial states and maintain long-term behavior despite temporal variations.

Contribution

It provides a detailed discussion of merging and stability properties in time inhomogeneous Markov chains, including examples illustrating long-term behavior under temporal variations.

Findings

Chains asymptotically forget initial states

Long-term behavior can be described by a binomial distribution

Temporal variations influence stability and merging properties

Abstract

We discuss problems posed by the quantitative study of time inhomogeneous Markov chains. The two main notions for our purpose are merging and stability. Merging (also called weak ergodicity) occurs when the chain asymptotically forgets where it started. It is a loss of memory property. Stability relates to the question of whether or not, despite temporary variations, there is a rough shape describing the long time behavior of the chain. For instance, we will discuss an example where the long time behavior is roughly described by a binomial, with temporal variations.