Implications of weak convergence rates of Markov transition kernels

TL;DR

This paper links weak convergence rates of Markov kernels to variance bounds for Lipschitz functions, aiding analysis of MCMC stability and stochastic algorithms.

Contribution

It extends weak convergence bounds to variance bounds and connects these to chi-squared divergence, providing new tools for MCMC and stochastic process analysis.

Findings

Weak convergence bounds imply variance bounds for Lipschitz functions.

In the reversible case, weak convergence implies chi-squared divergence bounds.

Applications include stability analysis of high-dimensional MCMC and stochastic algorithms.

Abstract

This article extends weak convergence bounds of Markov transition kernels to convergence bounds on the variance of the Markov kernel applied to Lipschitz functions. In the reversible case, weak convergence rates of the transition kernels imply chi-squared divergence convergence bounds if the density of the initialization measure is Lipschitz. These results provide new tools to establish central limit theorems for Lipschitz functions used in Markov chain Monte Carlo simulations. Applications are explored to the stability of Metropolis-Hastings algorithms in high dimensions, stochastic gradient descent, and solutions to stochastic delay equations.