On the distribution of the maximum of sums of mutually independent and identically distributed random variables

Throughout this paper we shall be concerned with a sequence of mutually independent and identically distributed random variables ξ1 ξ2, · · ·, ξ n , · · · taking on real values. We shall use the notation ζ n = ξ1 + · · · + ξ n for n = 1, 2, · · · and ζ0 = 0.