Almost Sure Central Limit Theorems and the Erdos-Renyi law for Expanding Maps of the Interval

Details Almost Sure Central Limit Theorems and the Erdos-Renyi law for Expanding Maps of the Interval Almost Sure Central Limit Theorems and the Erdos-Renyi law for Expanding Maps of the Interval

2002-10-18

arxiv.org/abs/math/0210286v2

Mathematics

Probability

25p; title has changed; minor corrections; published in Erg. Th. Dynam. Sys. (2005)

Scientific paper

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an exponential inequality recently obtained. Then we derive almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdos-Renyi type law). We end up with an application to entropy estimation ASCLT's that refi ne Shannon-McMillan-Breiman and Ornstein-Weiss theorems.

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