The Annals of Probability

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

Allan Gut and Svante Janson

Full-text: Open access

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
links.jstor.org

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J15 60F25: $L^p$-limit theorems 60K05: Renewal theory

Keywords
Random walk stopping time stopped random walk moments uniform integrability

Citation

Gut, Allan; Janson, Svante. Converse Results for Existence of Moments and Uniform Integrability for Stopped Random Walks. Ann. Probab. 14 (1986), no. 4, 1296--1317. doi:10.1214/aop/1176992371. https://projecteuclid.org/euclid.aop/1176992371


Export citation