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

Let 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 and , where 1[less-than-or-equals, slant]k=k(n)[less-than-or-equals, slant]n is an integer sequence that k(n)[+45 degree rule]n--> 0 and lim infn --> [infinity]k(n)[+45 degree rule]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).