A NOTE ON THE CONVERGENCE OF SUMS OF INDEPENDENT RANDOM-VARIABLES

Let X(n), n greater-than-or-equal-to 1, be a sequence of independent random variables, and let F(N) be the distribution function of the partial sums SIGMA(n = 1)N X(n). Motivated by a conjecture of Erdos in probabilistic number theory, we investigate conditions under which the convergence of F(N)(x) at two points x = x1, x2 with different limit values already implies the weak convergence of the distributions F(N). We show that this is the case if SIGMA(n = 1)infinity rho(X(n), c(n)) = infinity whenever SIGMA(n = 1)infinity c(n) diverges, where rho(X, c) denotes the Levy distance between X and the constant random variable c. In particular, this condition is satisfied if lim inf(n --> infinity) P(X(n) = 0) > 0.