Extremes of Shepp statistics for Gaussian random walk

Let (ξi, i ≥ 1) be a sequence of independent standard normal random variables and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_k=\sum\limits_{i=1}^{k}\xi_i$\end{document} be the corresponding random walk. We study the renormalized Shepp statistic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_T^{(N)}=\frac{1}{\sqrt{N}}\max\limits_{1\leq k\leq TN}\max\limits_{1\leq L\leq N}(S_{k+L-1}-S_{k-1})$\end{document} and determine asymptotic expressions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\textbf{\textrm{P}}\left(M_T^{(N)}>u\right)$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_T^{(N)}$\end{document} when T,N→ ∞ and present corresponding normalization sequences.