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April, 1973 Asymptotics of Randomly Stopped Sequences with Independent Increments
Priscilla Greenwood
Ann. Probab. 1(2): 317-321 (April, 1973). DOI: 10.1214/aop/1176996984


Let $S_n, n = 1, 2, \cdots$, be a sequence of sums of independent, identically distributed random variables $X_i$ such that $P\{X_i > y\}$ is a regularly varying function of $y$ at infinity. Let $N$ be a stopping time for $S_n$ with finite mean. A necessary and sufficient condition is given that $$\lim_{y\rightarrow \infty} P\{S_N > y\}/P\{X_1 > y\} = EN.$$ Examples further illustrate the role of this condition.


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Priscilla Greenwood. "Asymptotics of Randomly Stopped Sequences with Independent Increments." Ann. Probab. 1 (2) 317 - 321, April, 1973.


Published: April, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0259.60019
MathSciNet: MR350846
Digital Object Identifier: 10.1214/aop/1176996984

Primary: 60G40
Secondary: 60G50

Keywords: asymptotics , Independent increments , Random sums , regular variation , stopping times

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 2 • April, 1973
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