Renewal theory and computable convergence rates for geometrically ergodic Markov chains

We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward a “small set.” The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the first-entrance–last-exit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory.