Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T20:25:04.972Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence Properties of Perturbed Markov Chains

Part of: Parametric inference Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Gareth O. Roberts ,
Jeffrey S. Rosenthal  and
Peter O. Schwartz
Gareth O. Roberts*
Affiliation:
University of Cambridge
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
Peter O. Schwartz*
Affiliation:
University of Toronto
*
∗Postal address: Statistical Laboratory, University of Cambridge, Cambridge CB2 1SB, UK. e-mail address: [email protected]
∗∗Postal address: Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 1A1. e-mail address: [email protected]
∗∗∗Postal address: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1. e-mail address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the question of which convergence properties of Markov chains are preserved under small perturbations. Properties considered include geometric ergodicity and rates of convergence. Perturbations considered include roundoff error from computer simulation. We are motivated primarily by interest in Markov chain Monte Carlo algorithms.

Keywords

Markov chainMonte Carloperturbationroundoff errorgeometric convergenceconvergence rate

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62F99: None of the above, but in this section 62M05: Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 1 - 11
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.Google Scholar
Athreya, K.B., and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 493–501.Google Scholar
Ethier, S.N., and Kurtz, T.G. (1986). Markov Processes, Characterization and Convergence. Wiley, New York.Google Scholar
Gelfand, A.E., and Smith, A.F. M. (1990). Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85, 398–409.Google Scholar
Glynn, P.W., and Meyn, S.P. (1996). A Lyapunov bound for solutions of Poisson's equation. Preprint. Stanford University.Google Scholar
Kushner, H.J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. MIT Press, Cambridge, MA.Google Scholar
Meyn, S.P., and Tweedie, R.L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Meyn, S.P., and Tweedie, R.L. (1994). Computable bounds for convergence rates of Markov chains. Ann. Appl. Prob. 4, 981–1011.Google Scholar
Neal, R.M. (1993). Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRG-TR-93-1. University of Toronto.Google Scholar
Nummelin, E. (1984). General Irreducible Markov Chains and Non-Negative Operators. Cambridge University Press, Cambridge.Google Scholar
Roberts, G.O., and Tweedie, R.L. (1994). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Research Report 94-9. University of Cambridge.Google Scholar
Rosenthal, J.S. (1996). Analysis of the Gibbs sampler for a model related to James–Stein estimators. Statist. Comput. 9, 269–275.Google Scholar
Rosenthal, J.S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 558–566.Google Scholar
Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Prob. Comput. 1, 351–370.Google Scholar
Smith, A.F. M., and Roberts, G.O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. B 55, 3–24.Google Scholar
Sokal, A.D. (1989). Monte Carlo methods in statistical mechanics: foundations and new algorithms. Dept. of Physics, New York University. Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne, Switzerland.Google Scholar