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On the local limit theorem for non-uniformly ergodic Markov chains

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Marc Séva
Marc Séva*
Affiliation:
Université de Bretagne Occidentale
*
∗Postal address: Département de Mathématiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu BP 452, 29275 Brest Cedex, France.
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Abstract

Using an approach similar to that of Guivarc'h and Hardy (1988), we show that the local limit theorem holds for a Markov chain on a countable state space, with non-uniform ergodicity, when the recurrence is fast enough. We present a detailed study of a typical example, the reflected random walk on the positive half-line with negative mean and finite exponential moment. The results can be extended to some random walks on ℕ.

Keywords

CENTRAL LIMIT THEOREMRANDOM WALKPERTURBATION METHODS

MSC classification

Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 52 - 62
Copyright
Copyright © Applied Probability Trust 1995 

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References

Chung, K. L. (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Dacunha-Castelle, D. and Duflo, M. (1983) Probabilités et statistiques, t. 2: Problèmes à temps mobile. Masson, Paris.Google Scholar
Dunford, N. and Schwartz, J. T. (1967) Linear Operators, Part I Interscience, New York.Google Scholar
Guivarc'H, Y. and Hardy, J. (1988) Théorèmes limites pour une classe de chaînes de Markov et application aux difféomorphismes d'Anosov. Ann. Inst. H. Poincaré 24, 73–98.Google Scholar
Ionescu Tulcea, C. T. and Marinescu, G. (1950) Théorie ergodique pour des classes d'opérations non complètement continues. Ann. Math. (2) 52, 140–147.CrossRefGoogle Scholar
Iosifescu, M. and Grigorescu, S. (1990) Dependence with Complete Connections and its Applications. Cambridge University Press.Google Scholar
Kolmogorov, A. N. (1949) A local limit theorem for classical Markov chains. Izv, Akad. Nauk. SSSR. Ser. Math. 13, 281–300.Google Scholar
Krengel, A. (1985) Ergodic Theorems. Walter de Gruyter, Berlin.Google Scholar
Nagaev, S. V. (1957) Some limit theorems for stationary Markov chains. Theor. Prob. Appl. 2, 378–406.CrossRefGoogle Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-Negative Operators. Cambridge University Press.Google Scholar
Rousseau-Egèle, J. (1983) Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. Ann. Prob 11, 772–788.CrossRefGoogle Scholar