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On Functional Versions of the Arc-Sine Law

Author

Listed:
  • István Berkes

    (Graz University of Technology)

  • Siegfried Hörmann

    (University of Utah)

  • Lajos Horváth

    (University of Utah)

Abstract

Let X 1,X 2,… be a sequence of random variables. Let S k =X 1+⋅⋅⋅+X k and assume that S k /b k converges in distribution for some numerical sequence (b k ). We study the weak convergence of the random processes {Λ n (z), z∈ℝ}, where $$\Lambda_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}I\left\{\frac{S_{k}}{b_{k}}\leq z\right\}.$$ We consider the same problem when the normalized partial sums S k /b k are replaced by other functionals of the sequence (X n ). In particular, we investigate the case of sample extremes in detail.

Suggested Citation

  • István Berkes & Siegfried Hörmann & Lajos Horváth, 2010. "On Functional Versions of the Arc-Sine Law," Journal of Theoretical Probability, Springer, vol. 23(1), pages 109-126, March.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:1:d:10.1007_s10959-008-0181-7
    DOI: 10.1007/s10959-008-0181-7
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    References listed on IDEAS

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    1. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, March.
    2. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
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