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

Dedecker, J.; Gouëzel, S.; Merlevède, F.

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 796-821.

On considère une classe de transformations dilatantes T de [0, 1] ayant un point fixe neutre, ainsi que les chaînes de Markov associées Yi, dont le noyau de transition est l'opérateur de Perron-Frobenius de T par rapport à l'unique mesure de probabilité T-invariante possédant une densité. On montre une loi du logarithme itéré bornée pour les sommes partielles de f○Ti, lorsque f appartient à une classe de fonctions non bornées. Pour la même classe, on montre un principe d'invariance fort pour les sommes partielles de f(Yi). Lorsqu'on élargit la classe de fonctions, jusqu'à inclure des fonctions f pour lesquelles les sommes partielles de f○Ti appartiennent au domaine d'attraction normal d'une loi stable d'indice p∈(1, 2), on montre que les vitesses de convergence dans la loi forte des grands nombres sont les même que dans le cas i.i.d. correspondant.

We consider a large class of piecewise expanding maps T of [0, 1] with a neutral fixed point, and their associated Markov chains Yi 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○Ti 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(Yi) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f○Ti may belong to the domain of normal attraction of a stable law of index p∈(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.

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