Commun. Math. Phys. 229 (2002) 57-71
Ian Melbourne and Andrew Török
Abstract
We give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established. As an application of this method, we obtain new results for time-one maps of hyperbolic flows.
In particular, using recent results of Dolgopyat, we prove that many classical limit theorems of probability theory, such as the central limit theorem, the law of the iterated logarithm, and approximation by Brownian motion (almost sure invariance principle), are typically valid for such time-one maps.
The central limit theorem for hyperbolic flows goes back to Ratner 1973 and is always valid, irrespective of mixing hypotheses. We give examples which demonstrate that the situation for time-one maps is more delicate than that for hyperbolic flows, illustrating the need for rapid mixing hypotheses.