LIMIT THEOREMS FOR SOME FUNCTIONALS WITH HEAVY TAILS OF A DISCRETE TIME MARKOV CHAIN

Patrick Cattiaux (1) , Mawaki Manou-Abi (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 Sn = n i=1 f (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 [13, 21, 16, 4] for stationary mixing sequences. Contrary to the usual L 2 framework studied in [6], 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.

Mots clés: log-Sobolev inequality; stable distributions; functional limit theorem; additive functionals; stable limit theorems; Markov chains

Domaines: Mathématiques [math]; Statistiques [math.ST]

Patrick Cattiaux, Mawaki Manou-Abi. LIMIT THEOREMS FOR SOME FUNCTIONALS WITH HEAVY TAILS OF A DISCRETE TIME MARKOV CHAIN. ESAIM: Probability and Statistics, In press, ESAIM-PS, 18, pp.468-482. ⟨10.1051/ps/2013043⟩. ⟨hal-02063824⟩