Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains

Jerome Dedecker (1) , Sébastien Gouëzel (2) , Florence Merlevede (3)

1 LSTA - Laboratoire de Statistique Théorique et Appliquée
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We consider a large class of piecewise expanding maps T of [0,1] with a neutral fixed point, and their associated Markov chain Y_i whose transition kernel is the Perron-Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f\circ T^i satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Y_i) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f\circ T^i may belong to the domain of normal attraction of a stable law of index p\in (1, 2), we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.

Mots clés: Intermittency; almost sure convergence; law of the iterated logarithm; strong invariance principle

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Jerome Dedecker, Sébastien Gouëzel, Florence Merlevede. Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2010, 46 (3), pp.796-821. ⟨10.1214/09-AIHP343⟩. ⟨hal-00402864⟩