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Almost-sure central limit theorems and the Erdös–Rényi law for expanding maps of the interval

Published online by Cambridge University Press:  02 February 2005

J.-R. CHAZOTTES  and
P. COLLET
J.-R. CHAZOTTES
Affiliation:
Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique, F-91128 Palaiseau Cedex, France (e-mail: jeanrene@cpht.polytechnique.fr, collet@cpht.polytechnique.fr)
P. COLLET
Affiliation:
Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique, F-91128 Palaiseau Cedex, France (e-mail: jeanrene@cpht.polytechnique.fr, collet@cpht.polytechnique.fr)
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Abstract

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 rate of convergence in the Kantorovich distance. Maxima of partial sums are also shown to obey an ASCLT. The key tool is an exponential inequality recently obtained. Then we establish (optimal) almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdös–Rényi-type law). This is done by refining the usual large-deviations estimates available for expanding maps of the interval. We end up with an application to entropy estimation ASCLTs that refine the Shannon–McMillan–Breiman and Ornstein–Weiss theorems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 2 , April 2005 , pp. 419 - 441
Copyright
2005 Cambridge University Press

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