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Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

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Abstract

We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L 2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.

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Authors and Affiliations

  1. IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042, Rennes, France

    Sébastien Gouëzel

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  1. Sébastien Gouëzel
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Correspondence to Sébastien Gouëzel.

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Gouëzel, S. Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps. Isr. J. Math. 180, 1–41 (2010). https://doi.org/10.1007/s11856-010-0092-z

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  • DOI: https://doi.org/10.1007/s11856-010-0092-z

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