Concentration inequalities for Markov processes via coupling

TL;DR

This paper develops new concentration inequalities for Markov processes using coupling methods, providing bounds on moments and Gaussian deviations for Lipschitz functions along Markov chain paths.

Contribution

It introduces a coupling-based approach to derive moment and Gaussian bounds for Lipschitz functions of Markov chains on general state spaces.

Findings

Variance inequality when first moment of coupling time exists

Higher moment bounds under (1+epsilon) moment condition

Polynomial and Gaussian concentration inequalities derived

Abstract

We obtain moment and Gaussian bounds for general Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order 1+epsilon of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1+epsilon is finite uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.