The almost sure invariance principle for unbounded functions of expanding maps

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

2011-08-26

arxiv.org/abs/1108.5292v2

Mathematics

Probability

Scientific paper

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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