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Le théorème limite central pour des sommes de Riesz-Raikov
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  • Published: December 1992

Le théorème limite central pour des sommes de Riesz-Raikov

  • Bernard Petit1 

Probability Theory and Related Fields volume 93, pages 407–438 (1992)Cite this article

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  • 15 Citations

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Summary

Let θ be an algebraic number greater than 1 andf a real 1-periodic function; ifF N denotes the random variable defined on [0, 1] byF N (t)\( = \frac{1}{{\sqrt N }}\sum\limits_0^{N - 1} {f(\theta ^n t)} \), it is proved here that under sufficiently broad assumptions onf: 1) the sequence\(\left\{ {\mathop \smallint \limits_0^1 F_N^2 (t) dt} \right\}\) converges to a finite σ2(σ≧0); 2) if σ>0, the sequence {F N } converges in law to\(\mathfrak{N}(0,\sigma ^2 )\). We give an explicit computation of σ with respect to θ and a characterisation of functions for which σ=0.

(Our results are also valid for almost every real θ>1).

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Authors and Affiliations

  1. Department de Mathématiques, Université de Bretagne Occidentale, 6, Avenue Victor Le Gorgeu, F-29287, Brest, France

    Bernard Petit

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  1. Bernard Petit
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Petit, B. Le théorème limite central pour des sommes de Riesz-Raikov. Probab. Th. Rel. Fields 93, 407–438 (1992). https://doi.org/10.1007/BF01192715

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  • Received: 21 February 1991

  • Revised: 05 February 1992

  • Issue Date: December 1992

  • DOI: https://doi.org/10.1007/BF01192715

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