Approximate verification of geometric ergodicity for multiple-step Metropolis transition kernels

Abstract

In many applications involving discrete time Markov chains, the autocorrelation between states corresponding to nearby time points is too high to use all of these states as part of an approximate random sample from a specified target distribution. In these situations, it is common to use the output of a thinned chain, where we take samples every $h$ steps, and $h$ is a positive integer, in order to reduce autocorrelation. In order to justify using central limit theorems in analyses based on the output of a thinned chain, it is necessary to show that the thinned chain is geometrically ergodic. A common way to do this is to show that the chain satisfies a minorization condition and an associated drift condition. In this manuscript, we extend previous results pertaining to one-step transition kernels to handle numerical estimation of minorization and drift coefficients for $h$-step transition kernels for Metropolis algorithms.