Quantitative convergence rates for reversible Markov chains via strong random times

TL;DR

This paper develops a quantitative framework for analyzing the convergence rates of reversible Markov chains using strong random times, improving existing bounds and providing explicit convergence estimates based on drift and minorization conditions.

Contribution

It introduces a general principle linking strong random times to convergence rates for reversible chains with nonnegative eigenvalues, extending Baxendale's results with explicit bounds.

Findings

Derived tighter convergence bounds than previous methods

Converted drift and minorization data into explicit quantitative estimates

Validated approach on a well-studied example

Abstract

Let $(X_t)$ be a discrete time Markov chain on a general state space. It is well-known that if $(X_t)$ is aperiodic and satisfies a drift and minorization condition, then it converges to its stationary distribution $\pi$ at an exponential rate. We consider the problem of computing upper bounds for the distance from stationarity in terms of the drift and minorization data. Baxendale showed that these bounds improve significantly if one assumes that $(X_t)$ is reversible with nonnegative eigenvalues (i.e. its transition kernel is a self-adjoint operator on $L^2(\pi)$ with spectrum contained in $[0,1]$). We identify this phenomenon as a special case of a general principle: for a reversible chain with nonnegative eigenvalues, any strong random time gives direct control over the convergence rate. We formulate this principle precisely and deduce from it a stronger version of Baxendale's result. Our approach is fully quantitative and allows us to convert drift and minorization data into explicit convergence bounds. We show that these bounds are tighter than those of Rosenthal and Baxendale when applied to a well-studied example.