The Coupling/Minorization/Drift Approach to Markov Chain Convergence Rates

TL;DR

This paper reviews methods for analyzing Markov chain convergence rates, focusing on coupling, minorization, and drift techniques, with practical examples and applications to MCMC algorithms.

Contribution

It provides a comprehensive overview of convergence bounds using coupling, minorization, and drift conditions, highlighting their practical implementation and significance.

Findings

Quantitative bounds on convergence rates derived from minorization and drift conditions

Illustrative examples demonstrating practical application of theoretical bounds

Enhanced understanding of Markov chain convergence analysis for MCMC algorithms

Abstract

This review paper provides an introduction of Markov chains and their convergence rates which is an important and interesting mathematical topic which also has important applications for very widely used Markov chain Monte Carlo (MCMC) algorithm. We first discuss eigenvalue analysis for Markov chains on finite state spaces. Then, using the coupling construction, we prove two quantitative bounds based on minorization condition and drift conditions, and provide descriptive and intuitive examples to showcase how these theorems can be implemented in practice. This paper is meant to provide a general overview of the subject and spark interest in new Markov chain research areas.