Computable Convergence Rates for Subgeometrically Ergodic Markov Chains

TL;DR

This paper provides quantitative bounds on the convergence rates of Harris recurrent Markov chains under sub-geometric ergodicity conditions, with applications to queueing theory and MCMC.

Contribution

It introduces explicit bounds for sub-geometric convergence of Markov chains under drift and minorisation conditions, including the monotone case without minimal elements.

Findings

Derived bounds for sub-geometric ergodicity.

Applied bounds to queueing and MCMC examples.

Extended results to monotone chains without minimal elements.

Abstract

In this paper, we give quantitative bounds on the $f$-total variation distance from convergence of an Harris recurrent Markov chain on an arbitrary under drift and minorisation conditions implying ergodicity at a sub-geometric rate. These bounds are then specialized to the stochastically monotone case, covering the case where there is no minimal reachable element. The results are illustrated on two examples from queueing theory and Markov Chain Monte Carlo.