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

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