Extremes of Shepp statistics for fractional Brownian motion

Define the incremental fractional Brownian field with parameter H ∈ (0, 1) by ZH(τ, s) = BH(s +τ)-BH(s), where BH(s) is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We firstly derive the exact tail asymptotics for the maximum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_H^* (T) = \max _{(\tau ,s) \in [a,b] \times [0,T]} Z_H (\tau ,s)/\tau ^H $\end{document} of the standardised fractional Brownian motion field, with any fixed 0 < a < b < ∞ and T > 0; and we, furthermore, extend the obtained result to the case that T is a positive random variable independent of {BH(s), s ⩾ 0}. As a by-product, we obtain the Gumbel limit law for MH* (T) as T → ∞.