A "Fatou Equation" for Randomly Stopped Variables

Abstract

Let $X_n$ be a sequence of random variables adapted to an increasing sequence of $\sigma$-fields. In this note, convergence properties of $EX_t$ are studied as $t\rightarrow\infty$ through the directed set of stopping variables. The analogue of the inequality in Fatou's Lemma turns out to be an equation, which strengthens Fatou's Lemma. These problems arise naturally in the theory of gambling.