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Extremes of Shepp statistics for Gaussian random walk

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Abstract

Let (ξ i , i ≥ 1) be a sequence of independent standard normal random variables and let \(S_k=\sum\limits_{i=1}^{k}\xi_i\) be the corresponding random walk. We study the renormalized Shepp statistic \(M_T^{(N)}=\frac{1}{\sqrt{N}}\max\limits_{1\leq k\leq TN}\max\limits_{1\leq L\leq N}(S_{k+L-1}-S_{k-1})\) and determine asymptotic expressions for \(\textbf{\textrm{P}}\left(M_T^{(N)}>u\right)\) when u,N and T→ ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of \(M_T^{(N)}\) when T,N→ ∞ and present corresponding normalization sequences.

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  1. Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, Göteborg, Sweden

    Dmitrii Zholud

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  1. Dmitrii Zholud
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Zholud, D. Extremes of Shepp statistics for Gaussian random walk. Extremes 12, 1–17 (2009). https://doi.org/10.1007/s10687-008-0065-3

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  • DOI: https://doi.org/10.1007/s10687-008-0065-3

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