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

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

Received: 19 December 2012
Revised: 17 May 2013

Abstract

Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (X_n,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional ${\mathit{S}}_{\mathit{n}}\mathrm{=}{\sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{n}}\mathit{f}\mathrm{\left(}{\mathit{X}}_{\mathit{i}}\mathrm{\right)}$ 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 $\mathrm{L}{\mathrm{2}}_{}^{}$ 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.