# Criteria for Borel-Cantelli lemmas with applications to Markov chains and dynamical systems

## TL;DR

This paper establishes criteria for the Borel-Cantelli lemmas in the context of stationary sequences, Markov chains, and dynamical systems, highlighting conditions for almost sure occurrence of events based on dependence and mixing properties.

## Contribution

It provides new conditions under which Borel-Cantelli properties hold for dependent sequences, including Markov chains and dynamical systems, with specific focus on mixing rates and dependence structures.

## Key findings

- Borel-Cantelli property holds if μ(lim sup A_n) > 0 for absolutely regular sequences.
- Nested A_k sets require a convergence rate of mixing coefficients.
- Results extend to weaker dependence notions, applicable to non-irreducible Markov chains and dynamical systems.

## Abstract

Let (X k) be a strictly stationary sequence of random variables with values in some Polish space E and common marginal $\mu$, and (A k) k>0 be a sequence of Borel sets in E. In this paper, we give some conditions on (X k) and (A k) under which the events {X k $\in$ A k } satisfy the Borel-Cantelli (or strong Borel-Cantelli) property. In particular we prove that, if $\mu$(lim sup n A n) > 0, the Borel-Cantelli property holds for any absolutely regular sequence. In case where the A k 's are nested, we show, on some examples, that a rate of convergence of the mixing coefficients is needed. Finally we give extensions of these results to weaker notions of dependence, yielding applications to non-irreducible Markov chains and dynamical systems.