On the number of positive sums of independent random variables

Introduction. In a recent paper 2 the authors have introduced a method for proving certain limit theorems of the theory of probability. The purpose of the present note is to apply this general method in order to establish the following: THEOREM. Let Xi, X^ be independent random variables each having mean 0 and variance 1 and such that the central limit theorem is applicable. Let s& = Xi+X 2 + * * +Xk &nd let N n denote the number of Sk's, l^k^n, which are positive. Then,