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).
Publié le : 1986-10-14
Classification:
Random walk,
stopping time,
stopped random walk,
moments,
uniform integrability,
60G50,
60G40,
60J15,
60F25,
60K05
@article{1176992371,
author = {Gut, Allan and Janson, Svante},
title = {Converse Results for Existence of Moments and Uniform Integrability for Stopped Random Walks},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 1296-1317},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992371}
}
Gut, Allan; Janson, Svante. Converse Results for Existence of Moments and Uniform Integrability for Stopped Random Walks. Ann. Probab., Tome 14 (1986) no. 4, pp. 1296-1317. http://gdmltest.u-ga.fr/item/1176992371/