Converse Results for Existence of Moments and Uniform Integrability for Stopped Random Walks

Abstract

Let $\{S_n, n \geq 1\}$ be a random walk and $N$ a stopping time. The Burkholder-Gundy-Davis inequalities for martingales can be used to give conditions on the moments of $N$ (and of $X = S_1$), which ensure the finiteness of the moments of the stopped random walk, $S_N$. We establish converses to these results, that is, we obtain conditions on the moments of the stopped random walk and $X$ or $N$ which imply the finiteness of the moments of $N$ or $X$. We also study one-sided versions of these problems and corresponding questions concerning uniform integrability (of families of stopping times and families of stopped random walks).