Some limit results on supremum of Shepp statistics for fractional Brownian motion

Abstract

Define the incremental fractional Brownian field ZH(τ, s) = BH(s + τ) − BH(s), where BH(s) is a standard fractional Brownian motion with Hurst parameter H ∈ (0, 1). In this paper, we first derive an exact asymptotic of distribution of the maximum \({M_H}\left( {{T_u}} \right) = {\sup _{r \in [0,1],s \in [0,x{T_u}]}}{Z_H}\left( {\tau ,s} \right)\), which holds uniformly for x ∈ [A,B] with A,B two positive constants. We apply the findings to analyse the tail asymptotic and limit theorem of MH(T) with a random index T. In the end, we also prove an almost sure limit theorem for the maximum M1/2(T) with non-random index T.