Abstract
The purpose of this article is to support the idea that “whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average”. We follow three different approaches: martingale methods, spectral methods and induction arguments. Our results apply, among others, to Axiom A maps or flows, to systems inducing a Gibbs–Markov map, and to the stadium billiard.
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Chazottes, JR., Gouëzel, S. On almost-sure versions of classical limit theorems for dynamical systems. Probab. Theory Relat. Fields 138, 195–234 (2007). https://doi.org/10.1007/s00440-006-0021-6
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DOI: https://doi.org/10.1007/s00440-006-0021-6