Variance bounding Markov chains.

We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, vari- ance bounding is equivalent to the existence of usual central limit theorems for all L 2 functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis-Hastings algorithms.

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

[2]  S. Varadhan,et al.  Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions , 1986 .

[3]  Jeffrey S. Rosenthal,et al.  RATES OF CONVERGENCE FOR EVERYWHERE-POSITIVE MARKOV , 1995 .

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

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

[6]  Olle Häggström,et al.  On Variance Conditions for Markov Chain CLTs , 2007 .

[7]  Galin L. Jones.  On the Markov chain central limit theorem , 2004, math/0409112.

[8]  Galin L. Jones,et al.  Sufficient burn-in for Gibbs samplers for a hierarchical random effects model , 2004, math/0406454.

[9]  Galin L. Jones,et al.  On the applicability of regenerative simulation in Markov chain Monte Carlo , 2002 .

[10]  A. Sokal,et al.  Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality , 1988 .

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

[12]  James P. Hobert,et al.  Norm Comparisons for Data Augmentation , 2007 .

[13]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[14]  R. Tweedie,et al.  Exponential Convergence of Langevin Diiusions and Their Discrete Approximations , 1997 .

[15]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[16]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

[17]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[18]  Hani Doss.  Discussion: Markov Chains for Exploring Posterior Distributions , 1994 .

[19]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[20]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[21]  L. Tierney.  Rejoinder: Markov Chains for Exploring Posterior Distributions , 1994 .

[22]  Antonietta Mira,et al.  Ordering and Improving the Performance of Monte Carlo Markov Chains , 2001 .

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

[24]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[25]  J. A. Fill.  Eigenvalue bounds on convergence to stationarity for nonreversible markov chains , 1991 .

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

[27]  Walter R. Gilks,et al.  Strategies for improving MCMC , 1995 .

[28]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[29]  Gareth O. Roberts,et al.  Markov‐chain monte carlo: Some practical implications of theoretical results , 1998 .

[30]  J. Rosenthal,et al.  Convergence of Slice Sampler Markov Chains , 1999 .

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

[32]  G. O. Roberts.  A note on acceptance rate criteria for CLTS for Metropolis–Hastings algorithms , 1999 .

[33]  R. Tweedie,et al.  Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms , 1996 .

[34]  C. Geyer.  On Non-reversible Markov Chains , 2000 .

[35]  J. Rosenthal,et al.  Rates of convergence for everywhere-positive Markov chains , 1995 .

[36]  Djc MacKay,et al.  Slice sampling - Discussion , 2003 .

[37]  Gareth O. Roberts,et al.  Examples of Adaptive MCMC , 2009 .

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

[39]  Radford M. Neal,et al.  ANALYSIS OF A NONREVERSIBLE MARKOV CHAIN SAMPLER , 2000 .

[40]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

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

[42]  L. Tierney.  A note on Metropolis-Hastings kernels for general state spaces , 1998 .

[43]  G. Roberts,et al.  Perfect slice samplers , 2001 .

[44]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .