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

Abstract

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 o 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 o 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. © 2010 Association des Publications de l'Institut Henri Poincaré.