Quantitative bounds on convergence of time-inhomogeneous Markov chains

Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981-1101], Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and Tweedie [Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert [Statist. Sci. 16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566] that concerns quantitative convergence rates for time-homogeneous Markov chains. Our extension allows us to consider f-total variation distance (instead of total variation) and time-inhomogeneous Markov chains. We apply our results to simulated annealing.

[1]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[2]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[3]  D. Cox,et al.  Asymptotic techniques for use in statistics , 1989 .

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

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

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

[7]  R. Tweedie,et al.  Rates of convergence of the Hastings and Metropolis algorithms , 1996 .

[8]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[9]  Galin L. Jones,et al.  Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo , 2001 .

[10]  L. A. Breyer,et al.  Convergence of simulated annealing using Foster-Lyapunov criteria , 2001, Journal of Applied Probability.

[11]  G. Fort,et al.  Contrôle explicite d'ergodicité de chaîne de Markov : applications à l'analyse de convergence de l'algorithme Monte-Carlo EM , 2001 .

[12]  Gareth O. Roberts,et al.  SMALL AND PSEUDO-SMALL SETS FOR MARKOV CHAINS , 2001 .

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

[14]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[15]  Marco Locatelli,et al.  Convergence and first hitting time of simulated annealing algorithms for continuous global optimization , 2001, Math. Methods Oper. Res..

[16]  M. Locatelli.  Simulated Annealing Algorithms for Continuous Global Optimization , 2002 .

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

[18]  D. Draper,et al.  Stochastic Optimization: a Review , 2002 .