2019/04/03 by Dedecker, Jérôme, Merlevède, Florence, Rio, Emmanuel
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1904.01850
Let (X k) be a strictly stationary sequence of random variables with values in some Polish space E and common marginal μ, 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 ∈ A k satisfy the Borel-Cantelli (or strong Borel-Cantelli) property. In particular we prove that, if μ(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.