The Annals of Probability

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).

Article information

Source
Ann. Probab., Volume 14, Number 4 (1986), 1296-1317.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992371

Digital Object Identifier
doi:10.1214/aop/1176992371

Mathematical Reviews number (MathSciNet)
MR866351

Zentralblatt MATH identifier
0607.60055

JSTOR