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

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 MH(Tu)=supτ∈[0,1],s∈[0,xTu]ZH(τ,s),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(τ) with a random index τ.In the end,we also prove an ahnost sure limit theorem for the maximum M1/2(T) with non-random index T.