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Strong Central Limit Theorem for isotropic random walks in \({\mathbb{R}^d}\)
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  • Published: 29 April 2010

Strong Central Limit Theorem for isotropic random walks in \({\mathbb{R}^d}\)

  • Piotr Graczyk1,
  • Jean-Jacques Loeb1 &
  • Tomasz Żak2 

Probability Theory and Related Fields volume 151, pages 153–172 (2011)Cite this article

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Abstract

We prove an optimal Gaussian upper bound for the densities of isotropic random walks on \({\mathbb{R}^d}\) in spherical case (d ≥ 2) and ball case (d ≥ 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if \({\tilde S_n}\) denotes the normalized random walk and Y the limiting Gaussian vector, then \({\mathbb{E} f(\tilde S_{n}) \rightarrow \mathbb{E} f(Y)}\) for all functions f integrable with respect to the law of Y. We call such result a “Strong CLT”. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.

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Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques LAREMA, Université d’Angers, 2 boulevard Lavoisier, 49045, Angers Cedex 01, France

    Piotr Graczyk & Jean-Jacques Loeb

  2. Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Tomasz Żak

Authors
  1. Piotr Graczyk
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  2. Jean-Jacques Loeb
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  3. Tomasz Żak
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Corresponding author

Correspondence to Piotr Graczyk.

Additional information

This research was partially supported by grants MNiSW N N201 373136 and ANR-09-BLAN-0084-01.

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Graczyk, P., Loeb, JJ. & Żak, T. Strong Central Limit Theorem for isotropic random walks in \({\mathbb{R}^d}\) . Probab. Theory Relat. Fields 151, 153–172 (2011). https://doi.org/10.1007/s00440-010-0295-6

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  • Received: 08 December 2009

  • Revised: 06 March 2010

  • Published: 29 April 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0295-6

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Keywords

  • Random walks
  • Central Limit Theorem
  • Gaussian estimates
  • Logarithmic Sobolev inequality

Mathematics Subject Classification (2000)

  • 60G50
  • 60F05
  • 60B10
  • 47D06
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