中文摘要：

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.

英文摘要：

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）. Inthis paper, we first derive an exact asymptotic of distribution of the maximum MH（Tu） =supτ∈[0,1],s∈[0,xτu] ZH（τ, s）, which holds uniformly for x ∈ [A, B] with A, B two positive con-stants. We apply the findings to analyse the tail asymptotic and limit theorem of MH （τ） witha random index τ. In the end, we also prove an almost sure limit theorem for the maximum M1/2（τ） with non-random index T.