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Chapitre D'ouvrage Année : 2022

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

Résumé

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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Dates et versions

hal-02088063 , version 1 (02-04-2019)

Identifiants

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Jérôme Dedecker, Florence Merlevède, Emmanuel Rio. Criteria for Borel-Cantelli lemmas with applications to Markov chains and dynamical systems. Séminaire de Probabilités LI, Lectures Notes in Math. 2301, Springer, pp.189-238, 2022, ⟨10.1007/978-3-030-96409-2\_7⟩. ⟨hal-02088063⟩
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