On the convergence of the product of independent random variables (Q1097571)

Let \(\{X_ k\}\) be an independent random sequence such that \(X_ k+1>0\) a.s., and \({\mathbb{E}}[X_ k]=0\) for every k. In this paper the author proves that the following are equivalent: (A) \(\sum_{k}X_ k\) converges in L 1. (C) \(\sum_{k}X_ k\) converges almost surely. (F) \(\pi_ k(1+X_ k)\) converges in L 1. As an application he gives a new criterion for the mutually absolute continuity of infinite product probability measures on the sequence space.