CLTs and asymptotic variance of time-sampled Markov chains

TL;DR

This paper establishes CLT conditions and derives a spectral formula for the asymptotic variance of time-sampled Markov chains, enabling efficiency comparisons of algorithms like Barker's and Metropolis.

Contribution

It provides new CLT conditions and a spectral formula for asymptotic variance specific to time-sampled Markov chains, enhancing analysis of Markov chain efficiency.

Findings

Derived CLT conditions for time-sampled Markov chains

Established a spectral formula for asymptotic variance

Compared efficiency of Barker's and Metropolis algorithms

Abstract

For a Markov transition kernel $P$ and a probability distribution $ \mu$ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel $P_{\mu} = \sum_k \mu(k)P^k.$ In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker's and Metropolis algorithms in terms of asymptotic variance.