Quantitative convergence rates for sub-geometric Markov chains

TL;DR

This paper derives explicit formulas for convergence constants in sub-geometric Markov chains, aiding the analysis of ergodicity and uniform bounds in algorithms involving Markov kernels, including some inhomogeneous cases.

Contribution

It provides explicit expressions for ergodicity constants based on drift and minorisation conditions, extending to certain inhomogeneous chains.

Findings

Explicit constants for sub-geometric ergodicity derived

Applicable to inhomogeneous Markov chains

Facilitates uniform bounds in Markov chain algorithms

Abstract

We provide explicit expressions for the constants involved in the characterisation of ergodicity of sub-geometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The result is fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our result accommodates also some classes of inhomogeneous chains.