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

Abstract

Let S n 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 W n= max 1⩽i⩽n−k max 1⩽j⩽k (S i+j−S i), V n= max 1⩽i⩽n−k min 1⩽j⩽k (k⧸j)(S i+j−S i) and T n= max 1⩽i⩽n−k (S i+k(i)−S i , where 1⩽ k= k( n)⩽ n is an integer sequence that k( n)⧸ n→ 0 and lim inf n → ∞ k( n)⧸log n>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).