The Annals of Applied Probability

Renewal theory and computable convergence rates for geometrically ergodic Markov chains

Peter H. Baxendale

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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.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 700-738.

First available in Project Euclid: 1 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K05: Renewal theory 65C05: Monte Carlo methods

Geometric ergodicity renewal theory reversible Markov chain Markov chain Monte Carlo Metropolis–Hastings algorithm spectral gap


Baxendale, Peter H. Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 (2005), no. 1B, 700--738. doi:10.1214/105051604000000710.

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