A strong pair correlation bound implies the CLT for Sinai Billiards

TL;DR

This paper demonstrates that a strong pair correlation bound is sufficient to establish the CLT for Sinai Billiards and Anosov diffeomorphisms, broadening the class of observables for which the CLT holds.

Contribution

It shows that a strong pair correlation bound alone implies the CLT for Sinai Billiards and extends this result to higher-dimensional Anosov systems, relaxing previous regularity assumptions.

Findings

A strong pair correlation bound implies the CLT for Sinai Billiards.

The result extends to Anosov diffeomorphisms in any dimension.

Weaker regularity assumptions are sufficient for the correlation bound.

Abstract

For Dynamical Systems, a strong bound on multiple correlations implies the Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is derived for dynamically Holder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension.