Variance bounding Markov chains

TL;DR

This paper introduces the concept of variance bounding for Markov chains, establishing its relation to geometric ergodicity, and demonstrates its implications for central limit theorems and Metropolis-Hastings algorithms.

Contribution

It defines variance bounding, explores its properties, and connects it to existing ergodicity concepts, with applications to MCMC algorithms.

Findings

Variance bounding is weaker than geometric ergodicity but closely related.

Variance bounding ensures the validity of CLTs for all $L^2$ functionals.

Variance bounding is preserved under the Peskun order.

Abstract

We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all $L^2$ functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis--Hastings algorithms.