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 Diskr. Mat., 2004, Volume 16, Issue 1, Pages 140–145 (Mi dm148)

On the probabilities of large deviations of the Shepp statistic

A. M. Kozlov

Abstract: We find the asymptotic behaviour of the probability of large deviations $\mathsf P(W_{L,L}\geq\theta L)$ of the Shepp statistic $W_{L,L}$ which is equal to the maximum of fluctuations of the random walk
$$S_n=\sum_{i=1}^n\xi_i$$
in the window of width $L$ moving in the interval $[1,2L]$ as $L\to\infty$ and $\theta$ is a constant. We assume that $\xi_1,\xi_2,\ldots$ are independent identically distributed random variables with non-lattice distribution satisfying the right-side Cramer condition. We show that the asymptotics are of the form $H_\theta L\mathsf P(S_l\geq\theta L)$, where $H_\theta$ is a constant depending on $\theta$.
This research was supported by the Russian Foundation for Basic Research, grant 01–0100–649.

DOI: https://doi.org/10.4213/dm148

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English version:
Discrete Mathematics and Applications, 2004, 14:2, 211–216

Bibliographic databases:

UDC: 519.2

Citation: A. M. Kozlov, “On the probabilities of large deviations of the Shepp statistic”, Diskr. Mat., 16:1 (2004), 140–145; Discrete Math. Appl., 14:2 (2004), 211–216

Citation in format AMSBIB
\Bibitem{Koz04} \by A.~M.~Kozlov \paper On the probabilities of large deviations of the Shepp statistic \jour Diskr. Mat. \yr 2004 \vol 16 \issue 1 \pages 140--145 \mathnet{http://mi.mathnet.ru/dm148} \crossref{https://doi.org/10.4213/dm148} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2069995} \zmath{https://zbmath.org/?q=an:1050.60025} \transl \jour Discrete Math. Appl. \yr 2004 \vol 14 \issue 2 \pages 211--216 \crossref{https://doi.org/10.1515/156939204872356} 

• http://mi.mathnet.ru/eng/dm148
• https://doi.org/10.4213/dm148
• http://mi.mathnet.ru/eng/dm/v16/i1/p140

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This publication is cited in the following articles:
1. Zholud D., “Extremes of Shepp statistics for Gaussian random walk”, Extremes, 12:1 (2009), 1–17
2. Tan ZhongQuan, Yang Yang, “Extremes of Shepp Statistics For Fractional Brownian Motion”, Sci. China-Math., 58:8 (2015), 1779–1794
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