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Distributions diophantiennes et théorème limite local sur
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  • Published: 09 October 2004

Distributions diophantiennes et théorème limite local sur

  • E. Breuillard1 

Probability Theory and Related Fields volume 132, pages 13–38 (2005)Cite this article

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Abstract.

We study the speed of convergence of nd/2∫fdμ*n in the local limit theorem on under very general conditions upon the function f and the distribution μ. We show that this speed is at least of order and we give a simple characterization (in diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough f. We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when f is no longer assumed integrable but only bounded and Lipschitz or Hölder. We finally give an application to equidistribution of random walks.

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Authors and Affiliations

  1. DMA, Ecole Normale Supérieure, 45, rue d’Ulm, Paris Ve, France

    E. Breuillard

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  1. E. Breuillard
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Correspondence to E. Breuillard.

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Breuillard, E. Distributions diophantiennes et théorème limite local sur . Probab. Theory Relat. Fields 132, 13–38 (2005). https://doi.org/10.1007/s00440-004-0388-1

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  • Received: 16 August 2003

  • Revised: 14 July 2004

  • Published: 09 October 2004

  • Issue Date: May 2005

  • DOI: https://doi.org/10.1007/s00440-004-0388-1

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Keywords

  • Stochastic Process
  • General Condition
  • Probability Theory
  • Limit Theorem
  • Mathematical Biology
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