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Lois fortes des grands nombres presque sûres pour les sommes de Riesz–Raikov
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  • Published: November 2000

Lois fortes des grands nombres presque sûres pour les sommes de Riesz–Raikov

English title: Almost sure versions of the Riesz–Raikov strong law of large numbers

  • Emmanuel Rio1 

Probability Theory and Related Fields volume 118, pages 342–348 (2000)Cite this article

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Abstract.

The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L 1(ℝ/ℤ), then the averages

converge to the mean value of ƒ over [0, 1] for almost every x in [0, 1]. In this paper we prove that, for ƒ in L 1(ℝ/ℤ), the averages A n aƒ(x) converge a.e. to the integral of ƒ over [0, 1] for almost every a > 1. Furthermore we obtain convergence rates in this strong law of large numbers.

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Authors and Affiliations

  1. URA n° 0743 CNRS, Université de Paris-Sud, Bât. 425, Mathématique, 91405 Orsay Cedex, France. e-mail: emmanuel.rio@math.u-psud.fr, , , , , , FR

    Emmanuel Rio

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  1. Emmanuel Rio
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Received: 1 March 1999 / Revised version: 20 October 1999 / Published online: 12 October 2000

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Rio, E. Lois fortes des grands nombres presque sûres pour les sommes de Riesz–Raikov . Probab Theory Relat Fields 118, 342–348 (2000). https://doi.org/10.1007/PL00008745

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  • Issue Date: November 2000

  • DOI: https://doi.org/10.1007/PL00008745

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Keywords

  • Convergence Rate
  • Obtain Convergence Rate
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