Limit laws for the modulus of continuity of the partial sum process and for the shepp statistic

AbstractLet Sn denote the partial sum of an i.i.d. sequence of centred random variables having a finite moment generating function ø in a neighbourhood of zero. In this paper, we establish strong and weak limit laws for Wn=max1⩽i⩽n−kmax1⩽j⩽k(Si+j−Si), Vn=max1⩽i⩽n−kmin1⩽j⩽k(k⧸j)(Si+j−Si) and Tn=max1⩽i⩽n−k(Si+k(i)−Si, where 1⩽k=k(n)⩽n is an integer sequence that k(n)⧸n→ 0 and lim infn → ∞k(n)⧸logn>0. Our results extend those of Deheuvels, Devroye and Lynch (1986), Deheuvels and Devroye (1987), Deheuvels and Steinebach (1987) and M.Csörgõ and Steinebach (1981)