Criteria for Borel-Cantelli lemmas with applications to 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 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.
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https://hal.archives-ouvertes.fr/hal-02088063
Contributor : Jérôme Dedecker <>
Submitted on : Tuesday, April 2, 2019 - 3:21:14 PM
Last modification on : Thursday, April 11, 2019 - 4:02:09 PM

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  • HAL Id : hal-02088063, version 1
  • ARXIV : 1904.01850

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Jérôme Dedecker, Florence Merlevède, Emmanuel Rio. Criteria for Borel-Cantelli lemmas with applications to Markov chains and dynamical systems. 2019. ⟨hal-02088063⟩

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