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Criteria for Borel-Cantelli Lemmas with Applications to Markov Chains and Dynamical Systems

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Séminaire de Probabilités LI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2301))

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Abstract

Let (Xk) be a strictly stationary sequence of random variables with values in some Polish space E and common marginal μ, and (Ak)k>0 be a sequence of Borel sets in E. In this paper, we give some conditions on (Xk) and (Ak) under which the events {Xk ∈ Ak} satisfy the Borel-Cantelli (or strong Borel-Cantelli) property. In particular we prove that, if μ(limsupnAn) > 0, the Borel-Cantelli property holds for any absolutely regular sequence. In case where the Ak’s are nested, we show, on some examples, that a rate of convergence of the mixing coefficients is needed. Moreover we give extensions of these results to weaker notions of dependence, yielding applications to non necessarily irreducible Markov chains and dynamical systems. Finally, we show that some of our results are optimal in some sense by considering the case of Harris recurrent Markov chains.

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Acknowledgements

The authors are grateful to the referee for his careful reading of the manuscript and for his valuable comments.

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Authors and Affiliations

  1. Université de Paris, Laboratoire MAP5, Paris, France

    Jérôme Dedecker

  2. Université Gustave Eiffel, Marne-La-Vallée, France

    Florence Merlevède

  3. Université de Versailles, Laboratoire de mathématiques, Versailles, France

    Emmanuel Rio

Authors
  1. Jérôme Dedecker
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  2. Florence Merlevède
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  3. Emmanuel Rio
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Corresponding author

Correspondence to Emmanuel Rio .

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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Alain Rouault

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Dedecker, J., Merlevède, F., Rio, E. (2022). Criteria for Borel-Cantelli Lemmas with Applications to Markov Chains and Dynamical Systems. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités LI. Lecture Notes in Mathematics(), vol 2301. Springer, Cham. https://doi.org/10.1007/978-3-030-96409-2_7

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