On Functional Versions of the Arc-Sine Law

István Berkes (), Siegfried Hörmann () and Lajos Horvath
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István Berkes: Graz University of Technology
Siegfried Hörmann: University of Utah

Journal of Theoretical Probability, 2010, vol. 23, issue 1, 109-126

Abstract: 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.

Keywords: Arc-sine law; Invariance principles; Partial sums; Extremes; 60F17; 60G17; 60G50; 60G52; 60G70 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10959-008-0181-7

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