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

Cattiaux, Patrick  ; Manou-Abi, Mawaki 

ESAIM: Probability and Statistics, Tome 18 (2014), pp. 468-482

Résumé

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 $S_n={\sum }_{i=1}^{n}f\left(X_i\right)$ S n = ∑ i = 1 n f ( X i ) 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^2 L 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.

Détail

DOI : 10.1051/ps/2013043

Classification : 60F05, 60F17, 60J05, 60E07
Keywords: Markov chains, stable limit theorems, stable distributions, log-Sobolev inequality, additive functionals, functional limit theorem