Let $S_n$ be a stochastic process with either discrete or continuous time parameter and stationary independent increments. Let $N$ be a stopping time for the process such that $EN < \infty$. If the upper tail of the process distribution, $F$, is regularly varying, certain conditions on the lower tail of $F$ and on the tail of the distribution of $N$ imply that $\lim_{y\rightarrow\infty}P(S_N > y)/(1 - F(y)) = EN$. A similar asymptotic relation is obtained for $\sup_n S_{n \wedge N}$, if $n$ is discrete. These asymptotic results are related to the Wald moment identities and to moment inequalities of Burkholder. Applications are given for exit times at fixed and square-root boundaries.
Publié le : 1977-02-14
Classification:
Stopping times,
random walk,
independent increments,
regular variation,
asymptotics,
boundary crossing,
maximum process,
60G40,
60J15,
60J30
@article{1176995889,
author = {Greenwood, Priscilla and Monroe, Itrel},
title = {Random Stopping Preserves Regular Variation of Process Distributions},
journal = {Ann. Probab.},
volume = {5},
number = {6},
year = {1977},
pages = { 42-51},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995889}
}
Greenwood, Priscilla; Monroe, Itrel. Random Stopping Preserves Regular Variation of Process Distributions. Ann. Probab., Tome 5 (1977) no. 6, pp. 42-51. http://gdmltest.u-ga.fr/item/1176995889/