A Borel-Cantelli lemma and its applications

TL;DR

This paper presents a new version of the Borel-Cantelli lemma and demonstrates its applications in proving an almost sure local central limit theorem and a dynamical Borel-Cantelli lemma for systems with rapid decay of correlations.

Contribution

It introduces a modified Borel-Cantelli lemma and applies it to establish results in probability theory and dynamical systems.

Findings

Proves an almost sure local central limit theorem.

Establishes a dynamical Borel-Cantelli lemma for systems with fast decay of correlations.

Provides a new version of the Borel-Cantelli lemma.

Abstract

We give a version of the Borel-Cantelli lemma. As an application, we prove an almost sure local central limit theorem. As another application, we prove a dynamical Borel-Cantelli lemma for systems with sufficiently fast decay of correlations with respect to Lipschitz observables.