On the central limit theorem for aperiodic dynamical systems and application. (English) Zbl 0938.37007

De la Rue, T.; Ladouceur, S.; Peskir, G.; Weber, M.

Theory Probab. Math. Stat. 57, 149-169 (1998) and Teor. Jmovirn. Mat. Stat. 57, 140-159 (1997).

The main purpose of this paper is to exhibit a real valued function \(f\) defined on the phase space \(X\) of a given aperiodic dynamical system \((X, {\mathcal A}, \mu, T)\) such that the natural long-term ratio satisfies the central limit theorem. The method of proving is based on Burton-Denker’s construction which in turn relies upon Kakutani-Rochlin’s lemma. A fundamental fact in the background of the entire construction is provided by using Rochlin’s result on a factor space of Lebesgue space. This result is extended to a more general case involving orbits of aperiodic dynamical systems.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

37C27	Periodic orbits of vector fields and flows
37H10	Generation, random and stochastic difference and differential equations
60F05	Central limit and other weak theorems

Keywords:

central limit theorem; aperiodic dynamical systems; Kakutani-Rochlin’s lemma; factor space; Barry-Esseen’s theorem; continuity theorem; orbit; Gaussian; GB-set; Lebesgue space; aperiodic