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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.
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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 

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