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The central limit theorem for Riesz-Raikov sums
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  • Published: March 1994

The central limit theorem for Riesz-Raikov sums

  • Katusi Fukuyama1 nAff2 

Probability Theory and Related Fields volume 100, pages 57–75 (1994)Cite this article

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Summary

Letf be a real-valued function with period 1 satisfying some regularity conditions. It will be proved that, for anyθ 1,...,θ d > 1, the distribution of\(n^{ - 1/2} \sum\nolimits_{k = 1}^n {\left( {f(\theta _1^k t),...,f(\theta _d^k t)} \right)} \) on the probability space ([0, 1],dt) converges to a normal distribution whose covariance is given by algebraic relations among theθ i 's. This generalizes the classical work by M. Kac and refines the characterization off to have a degenerate limit. It also shows that the limit law of\(\sum\nolimits_{k = 1}^n {f(\theta _1^k t)/\sum\nolimits_{l = 1}^n {f(\theta _j^l t)} } \) is in most cases a Cauchy distribution.

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Author notes
  1. Katusi Fukuyama

    Present address: Department of Mathematics, Kobe University, Rokko, 657, Kobe, Japan

Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

    Katusi Fukuyama

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  1. Katusi Fukuyama
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Fukuyama, K. The central limit theorem for Riesz-Raikov sums. Probab. Th. Rel. Fields 100, 57–75 (1994). https://doi.org/10.1007/BF01204953

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  • Received: 18 January 1994

  • Revised: 07 March 1994

  • Issue Date: March 1994

  • DOI: https://doi.org/10.1007/BF01204953

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Mathematics Subject Classification (1991)

  • 60F05
  • 42A55
  • 11K70
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