A Generalization of the Hsu-Robbins-Erdős Theorem

In case (Xn) is an i.i.d. sequence, and Sn = Xl + ••• + Xn, the Hsu-Robbins-Erdős theorem states that Σxn=1P(|Sn| > nϵ) < ∞, ϵ > 0, iff EX21 < ∞ and EX1 = 0. The purpose of this paper is to generalize this result to the case in which the steps Xn are independent, but their distributions are taken from a finite set.