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 dierent from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition towards 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.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  D. Vere-Jones.  On the spectra of some linear operators associated with queueing systems , 1963 .

[3]  R. Tweedie,et al.  Geometric Ergodicity and R-positivity for General Markov Chains , 1978 .

[4]  E. Nummelin,et al.  Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory , 1982 .

[5]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Embedded renewal processes , 1984 .

[6]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[7]  S. Meyn,et al.  Computable Bounds for Geometric Convergence Rates of Markov Chains , 1994 .

[8]  J. Rosenthal.  Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo , 1995 .

[9]  Jun S. Liu,et al.  Covariance Structure and Convergence Rate of the Gibbs Sampler with Various Scans , 1995 .

[10]  R. L. Tweedie,et al.  Explicit Rates of Convergence of Stochastically Ordered Markov Chains , 1996 .

[11]  Richard L. Tweedie,et al.  Geometric Convergence Rates for Stochastically Ordered Markov Chains , 1996, Math. Oper. Res..

[12]  J. Rosenthal,et al.  Shift-coupling and convergence rates of ergodic averages , 1997 .

[13]  J. Rosenthal,et al.  Geometric Ergodicity and Hybrid Markov Chains , 1997 .

[14]  R. L. Tweedie,et al.  Rates of convergence of stochastically monotone and continuous time Markov models , 2000 .

[15]  R. Tweedie,et al.  Geometric L 2 and L 1 convergence are equivalent for reversible Markov chains , 2001, Journal of Applied Probability.

[16]  Gareth O. Roberts,et al.  Corrigendum to : Bounds on regeneration times and convergence rates for Markov chains , 2001 .

[17]  S. Rosenthal,et al.  Asymptotic Variance and Convergence Rates of Nearly-Periodic MCMC Algorithms , 2002 .

[18]  J. Rosenthal.  QUANTITATIVE CONVERGENCE RATES OF MARKOV CHAINS: A SIMPLE ACCOUNT , 2002 .

[19]  Computable bounds for V-geometric ergodicity of Markov transition kernels , 2003 .

[20]  R. Douc,et al.  Quantitative bounds on convergence of time-inhomogeneous Markov chains , 2004, math/0503532.