Asymptotic variance of stationary reversible and normal Markov processes

TL;DR

This paper establishes conditions for the variance growth of functionals of stationary Markov processes and constructs Metropolis-Hastings algorithms satisfying a CLT even when variance growth is non-linear.

Contribution

It provides necessary and sufficient conditions for variance regular variation and introduces Metropolis-Hastings algorithms with CLT validity under non-linear variance growth.

Findings

Variance of partial sums exhibits regular variation under certain conditions.

Constructed Metropolis-Hastings algorithms satisfy CLT and invariance principle with non-linear variance.

Conditions for normal transition operators to ensure variance behavior.

Abstract

We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in $n$.