Which ergodic averages have finite asymptotic variance?

TL;DR

This paper characterizes which ergodic averages of reversible Markov chains have finite asymptotic variance, providing a complete description for independence samplers and a sufficient condition for pseudo-marginal chains.

Contribution

It offers a complete characterization of finite asymptotic variance for ergodic averages in reversible Markov chains, especially for independence samplers and pseudo-marginal chains.

Findings

Characterization of $L^2$ functions with finite asymptotic variance.

Complete description for independence sampler ergodic averages.

Sufficient condition for pseudo-marginal Markov chains.

Abstract

We show that the class of $L^2$ functions for which ergodic averages of a reversible Markov chain have finite asymptotic variance is determined by the class of $L^2$ functions for which ergodic averages of its associated jump chain have finite asymptotic variance. This allows us to characterize completely which ergodic averages have finite asymptotic variance when the Markov chain is an independence sampler. In addition, we obtain a simple sufficient condition for all ergodic averages of $L^2$ functions of the primary variable in a pseudo-marginal Markov chain to have finite asymptotic variance.