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A sufficient condition for the existence of an invariant probability measure for Markov processes

Part of: Stochastic systems and control Markov processes

Published online by Cambridge University Press:  14 July 2016

O. L. V. Costa  and
F. Dufour
O. L. V. Costa*
Affiliation:
Universidade de São Paulo
F. Dufour*
Affiliation:
Université Bordeaux I and Université Bordeaux IV
*
∗Postal address: Departamento de Engenharia de Telecomunicações e Controle, Escola Politécnica da Universidade de São Paulo, São Paulo, 05508 900, Brazil. Email address: oswaldo@lac.usp.br
∗∗Postal address: Mathématiques Appliqées de Bordeaux, Université Bordeaux I, 351 cours de la Liberation, 33405 Talence Cedex, France. Email address: dufour@math.u-bordeaux1.fr
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Abstract

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In this paper, it is shown that the Foster-Lyapunov criterion is sufficient to ensure the existence of an invariant probability measure for both discrete- and continuous-time Markov processes without any additional hypotheses (such as irreducibility).

Keywords

Markov processinvariant probability measureFoster-Lyapunov criterionpetite setdiscrete timecontinuous time

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 93E15: Stochastic stability
Type
Short Communications
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 873 - 878
Copyright
© Applied Probability Trust 2005 

References

Azéma, J., Kaplan-Duflo, M. and Revuz, D. (1967). Mesure invariante sur les classes récurrentes des processus de Markov. Z. Wahrscheinlichkeitsth. 8, 157–181.CrossRefGoogle Scholar
Costa, O. L. V. and Dufour, F. (2005). On the ergodic decomposition for a class of Markov chains. Stoch. Process. Appl. 115, 401–415.Google Scholar
Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23, 1671–1691.Google Scholar
Lasserre, J. B. (1997). Invariant probabilities for Markov chains on a metric space. Statist. Prob. Lett. 34, 259–266.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. Appl. Prob. 24, 542–574.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.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. (1993). {Markov Chains and Stochastic Stability}. Springer, Berlin.Google Scholar
Tweedie, R. L. (1976). Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
Tweedie, R. L. (1988). Invariant measures for {Markov chains with no irreducibility assumptions.} In A Celebration of Applied Probability (J. Appl. Prob. Spec. Vol. 25A), Applied Probability Trust, Sheffield, pp. 275–285.Google Scholar
Tweedie, R. L. (2001). Drift conditions and invariant measures for Markov chains. Stoch. Process. Appl. 92, 345–354.Google Scholar