A Note on the Convergence of Sums of Independent Random Variables

Abstract

Let $X_n, n \geq 1$, be a sequence of independent random variables, and let $F_N$ be the distribution function of the partial sums $\sum^N_{n = 1}X_n$. Motivated by a conjecture of Erdos in probabilistic number theory, we investigate conditions under which the convergence of $F_N(x)$ at two points $x = x_1,x_2$ with different limit values already implies the weak convergence of the distributions $F_N$. We show that this is the case if $\sum^\infty_{n = 1}\rho(X_n,c_n) = \infty$ whenever $\sum^\infty_{n = 1}c_n$ diverges, where $\rho(X,c)$ denotes the Levy distance between $X$ and the constant random variable $c$. In particular, this condition is satisfied if $\lim\inf_{n \rightarrow\infty}P(X_n = 0) > 0$.