Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances

TL;DR

This paper develops explicit methods to bound the convergence rates of Markov chains in total variation and Wasserstein distances, using Steinsaltz's theorem and one-shot coupling, with practical applications to Gibbs sampling and logistic systems.

Contribution

It introduces a unified framework for deriving explicit convergence bounds for Markov chains in both distances, utilizing local contractivity and coupling techniques.

Findings

Derived practical criteria for Wasserstein convergence bounds.

Established total variation bounds via Wasserstein distances.

Applied methods to Gibbs sampler and logistic dynamical system.

Abstract

We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool is Steinsaltz's convergence theorem for locally contractive random dynamical systems. We describe practical methods for finding Steinsaltz's "drift functions" that prove local contractivity. We then use the idea of "one-shot coupling" to derive criteria that give bounds for total variation distances in terms of Wasserstein distances. Our methods are applied to two examples: a two-component Gibbs sampler for the Normal distribution and a random logistic dynamical system.