Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-11T16:22:12.511Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgeometric Rates of Convergence of f-Ergodic Markov Chains

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Pekka Tuominen  and
Richard L. Tweedie
Pekka Tuominen*
Affiliation:
University of Helsinki
Richard L. Tweedie*
Affiliation:
Colorado State University
*
* Postal address: Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00100 Helsinki, Finland.
** Postal address: Department of Statistics, Colorado State University, Fort Collins CO 80523, USA. Work supported in part by NSF Grant DMS-9205687 and Academy of Finland Grant 1011733.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Φ = {Φ n} be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of Ex[g(Φ n)] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which

as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of

(i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and

(ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions).

The results are illustrated for random walks and for more general state space models.

Keywords

IRREDUCIBLE MARKOV PROCESSESERGODICITYHARRIS RECURRENCEINVARIANT MEASURESGEOMETRIC ERGODICITYRANDOM WALKSTATE SPACE MODELS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K05: Renewal theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 775 - 798
Copyright
Copyright © Applied Probability Trust 1994 

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

[1] Lindvall, T. (1979) On coupling of discrete renewal processes. Z. Wahrscheinlichkeitsth. 48, 57–70.CrossRefGoogle Scholar
[2] Meyn, S. P. and Tweedie, R. L. (1992) Stability of Markovian processes I: Discrete time chains. Adv. Appl. Prob. 24, 542–574.CrossRefGoogle Scholar
[3] Meyn, S. P. and Tweedie, R. L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag, London.CrossRefGoogle Scholar
[4] Nummelin, E. (1984) General Irreducible Markov Chains and Non-Negative Operators. Cambridge University Press.CrossRefGoogle Scholar
[5] Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Proc. Appl. 12, 187–202.CrossRefGoogle Scholar
[6] Nummelin, E. and Tuominen, P. (1983) The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory. Stoch. Proc. Appl. 15, 295–311.CrossRefGoogle Scholar
[7] Nummelin, E. and Tweedie, R. L. (1978) Geometric ergodicity and R-positivity for general Markov chains. Ann. Prob. 6, 404–420.CrossRefGoogle Scholar
[8] Orey, S. (1971) Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
[9] Stone, C. and Wainger, S. (1967) One-sided error estimates in renewal theory. J. d'Anal. Math. XX, 325–352.CrossRefGoogle Scholar
[10] Thorisson, H. (1983) The coupling of regenerative process. Adv. Appl. Prob. 15, 531–561.CrossRefGoogle Scholar
[11] Thorisson, H. (1985) On regenerative properties of the k-server queue with non-stationary Poisson arrivals. J. Appl. Prob. 22, 893–902.CrossRefGoogle Scholar
[12] Thorisson, H. (1985) The queue GI//G/1: finite moments of the cycle variables and uniform rates of convergence. Stoch. Proc. Appl. 19, 85–99.CrossRefGoogle Scholar
[13] Tjøstheim, D. (1990) Non-linear time series and Markov chains. Adv. Appl. Prob. 22, 587–611.CrossRefGoogle Scholar
[14] Tweedi, R. L. (1983) Criteria for rates of convergence of Markov chains with application to queueing and storage theory. In Probability Statistics and Analysis. Ed. Kingman, J. F. C. and Reuter, G. E. H., London Mathematics Society Lecture Note Series, Cambridge University Press.Google Scholar
[15] Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar