Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:49:23.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Rates of convergence of stochastically monotone and continuous time Markov models

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

G. O. Roberts  and
R. L. Tweedie
G. O. Roberts*
Affiliation:
University of Lancaster
R. L. Tweedie*
Affiliation:
School of Public Health, Minneapolis
*
∗Postal address: Department of Mathematics and Statistics, University of Lancaster, Lancaster LA1 4YF, England. Email address: g.o.roberts@lancaseter.ac.uk.
∗∗Postal address: Division of Biostatistics, University of Minnesota, A460 Mayo Building, Box 303, 420 Delaware Street, SE Minneapolis, MN 55455-0378, USA. Email address: tweedie@bistat.umn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we give bounds on the total variation distance from convergence of a continuous time positive recurrent Markov process on an arbitrary state space, based on Foster-Lyapunov drift and minorisation conditions. Considerably improved bounds are given in the stochastically monotone case, for both discrete and continuous time models, even in the absence of a reachable minimal element. These results are applied to storage models and to diffusion processes.

Keywords

Stochastic monotonicityrates of convergenceMarkov chainMarkov process

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 359 - 373
Copyright
Copyright © by the Applied Probability Trust 2000 

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.)

Footnotes

Work supported in part by NSF Grant DMS 9803682 and EPSRC grant GR/J19900.

References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, New York.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Liu, J. (1996). Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statist. Computing 6, 113–119.Google Scholar
Lund, R. B., and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 21, 182–194.Google Scholar
Lund, R. B., Meyn, S. P., and Tweedie, R. L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6, 218–237.Google Scholar
Kalashnikov, V. K. (1994). Regeneration and general Markov chains. J. Applied Math. Stochast. Anal. 7, 357–371.Google Scholar
Kalashnikov, V. K. (1994). Topics on Regenerative Processes. CRC Press, Boca Raton.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer-Verlag, London.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous time processes. Adv. Appl. Prob. 25, 518–548.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
Roberts, G. O., and Rosenthal, J. S. (1996). Quantitative bounds for convergence rates of continuous time Markov process. Electr. J. Prob. 1, Paper 9, pp. 1-21.Google Scholar
Roberts, G. O., and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates of Markov chains. Stoch. Proc Appl. 80, 211–229.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
Rosenthal, J. S. (1996). Convergence of Gibbs sampler for a model related to James–Stein estimators. Statist. Comput. 6, 269–275.Google Scholar
Scott, D. J., and Tweedie, R. L. (1996). Explicit rates of convergence of stochastically ordered Markov chains. In Proc. Athens Conf. on Applied Probability and Time Series Analysis: Papers in Honour of J. M. Gani and E. J. Hannan. Springer, New York, pp. 176–191.Google Scholar
Smith, R. L., and Tierney, L. (1997). Exact transition probabilities for the independence Metropolis sampler. To appear in Bernoulli. (Preprint available at http://www.stats.bris.ac.uk/MCMC/.)Google Scholar