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Limit theorems for some functionals with heavy tails of a discrete time Markov chain

Published online by Cambridge University Press:  08 October 2014

Patrick Cattiaux  and
Mawaki Manou-Abi
Patrick Cattiaux
Affiliation:
Institut de Mathématiques de Toulouse. CNRS UMR 5219. Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 09, France. cattiaux@math.univ-toulouse.fr; manoumawaki@gmail.com
Mawaki Manou-Abi
Affiliation:
Institut de Mathématiques de Toulouse. CNRS UMR 5219. Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 09, France. cattiaux@math.univ-toulouse.fr; manoumawaki@gmail.com
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Abstract

Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional \hbox{$S_{n}=\sum_{i=1}^{n}f(X_{i})$}Sn=∑i=1nf(Xi) for a possibly non square integrable function f. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab. Lett. 8 (1989) 477–483; M. Tyran-Kaminska, Stochastic Process. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis (2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40 (2012) 2008–2033] for stationary mixing sequences. Contrary to the usual L^2L2 framework studied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2012) 337–382], where weak forms of ergodicity are sufficient to ensure the validity of the Central Limit Theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally we give explicit examples.

Keywords

Markov chainsstable limit theoremsstable distributionslog-Sobolev inequalityadditive functionalsfunctional limit theorem
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 468 - 482
Copyright
© EDP Sciences, SMAI 2014

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