The almost sure invariance principle for unbounded functions of expanding maps

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

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps with a neutral fixed point at zero. In both cases, we give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy an almost sure invariance principle. This class contains piecewise monotonic functions (with a finite number of branches) such that: - For uniformly expanding maps, they are square integrable with respect to the absolutely continuous invariant probability measure. - For maps having a neutral fixed point at zero, they satisfy an (optimal) tail condition with respect to the absolutely continuous invariant probability measure.

Mots clés: Expanding maps; intermittency; strong invariance principle

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

Jerome Dedecker, Sébastien Gouëzel, Florence Merlevede. The almost sure invariance principle for unbounded functions of expanding maps. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2012, 9, pp.141-163. ⟨hal-00617189v2⟩