Convergence of sum product of a martingale difference sequence (Q1113179)

The author proves the following theorem: Let \(\{X_ k,\beta_ k\}\) be a martingale difference sequence such that \(X_ k+1>0\) a.s., for every \(k\geq 1\). Then \(\Pi_ k(1+X_ k)\) converges almost surely if and only if \(\sum_{k}X_ k\) converges almost surely. As an application of this theorem the author gives a new criterion for the absolute continuity of locally equivalent measures.