Extremes of Shepp statistics for the Wiener process

Journal article, 2008

Define Y(t)=max0≤s≤1W(t+s)−W(t)Y(t)=max0≤s≤1W(t+s)−W(t) , where W(·) is a standard Wiener process. We study the maximum of Y up to time T: MT=max0≤t≤TY(t)MT=max0≤t≤TY(t) and de termine an asymptotic expression for P(MT>u)P(MT>u) when u→ ∞. Further we establish the limiting Gumbel distribution of M T when T→ ∞ and present the corresponding normalization sequence.