Perturbation theory and uniform ergodicity for discrete-time Markov chains

TL;DR

This paper investigates perturbation theory and uniform ergodicity for discrete-time Markov chains on general state spaces, introducing new methods based on uniform moments of first hitting times to analyze convergence rates.

Contribution

It develops novel techniques linking geometric moments of first return times to convergence rates, especially for reversible Markov chains, and extends analysis to the two-skeleton chain.

Findings

Established new bounds for convergence rates using uniform moments.

Connected geometric moments of return times between P and P^2.

Provided insights into ergodicity for reversible and non-reversible chains.

Abstract

We study perturbation theory and uniform ergodicity for discrete-time Markov chains on general state spaces in terms of the uniform moments of the first hitting times on some set. The methods we adopt are different from previous ones. For reversible and non-negative definite Markov chains, we first investigate the geometrically ergodic convergence rates. Based on the estimates, together with a first passage formula, we then get the convergence rates in uniform ergodicity. If the transition kernel $P$ is only reversible, we transfer to study the two-skeleton chain with the transition kernel $P^2$. At a technical level, the crucial point is to connect the geometric moments of the first return times between $P$ and $P^2$.