Open Access
February, 1977 Random Stopping Preserves Regular Variation of Process Distributions
Priscilla Greenwood, Itrel Monroe
Ann. Probab. 5(1): 42-51 (February, 1977). DOI: 10.1214/aop/1176995889


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.


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Priscilla Greenwood. Itrel Monroe. "Random Stopping Preserves Regular Variation of Process Distributions." Ann. Probab. 5 (1) 42 - 51, February, 1977.


Published: February, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0379.60043
MathSciNet: MR426139
Digital Object Identifier: 10.1214/aop/1176995889

Primary: 60G40
Secondary: 60J15 , 60J30

Keywords: asymptotics , Boundary crossing , Independent increments , maximum process , Random walk , regular variation , stopping times

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 1 • February, 1977
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