# The almost sure invariance principle for unbounded functions of expanding maps

Abstract : 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.
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Type de document :
Article dans une revue
ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2012, 9, pp.141-163
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https://hal.archives-ouvertes.fr/hal-00617189
Contributeur : Sebastien Gouezel <>
Soumis le : mercredi 25 janvier 2012 - 16:23:34
Dernière modification le : vendredi 24 février 2017 - 01:02:01
Document(s) archivé(s) le : jeudi 26 avril 2012 - 02:31:44

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### Identifiants

• HAL Id : hal-00617189, version 2
• ARXIV : 1108.5292

### Citation

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, Instituto Nacional de Matemática Pura e Aplicada, 2012, 9, pp.141-163. <hal-00617189v2>

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