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October, 1974 On Stopping Rules and the Expected Supremum of $S_n/a_n$ and $|S_n|/a_n$
Michael J. Klass
Ann. Probab. 2(5): 889-905 (October, 1974). DOI: 10.1214/aop/1176996555


Let $\{X_n\}$ be a sequence of i.i.d. mean zero random variables. Let $S_n = X_1 + \cdots + X_n$. This paper is devoted to determining the conditions where-by $E\sup_{n\geqq 1}S_n/a_n < \infty$ and $E\sup_{n\geqq 1}|S_n|/a_n < \infty$ for quite general sequences of increasing constants $\{a_n\}$. For the sequences $\{a_n\}$ considered, we find it sufficient to examine whether or not $\lim_{n\rightarrow\infty} E(\sum^n_{k=1}X_k/a_k)^+ < \infty$. The existence of optimal extended-valued stopping rules with finite expected reward for sequences $\{S_n/a_n\}$ or $\{|S_n|/a_n\}$ is a by-product of our results. This generalizes results of D. L. Burkholder, Burgess Davis, R. F. Gundy, B. J. McCabe and L. A. Shepp, who treat the case $a_n = n$.


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Michael J. Klass. "On Stopping Rules and the Expected Supremum of $S_n/a_n$ and $|S_n|/a_n$." Ann. Probab. 2 (5) 889 - 905, October, 1974.


Published: October, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0325.60043
MathSciNet: MR380967
Digital Object Identifier: 10.1214/aop/1176996555

Primary: 60G40
Secondary: 60G50

Keywords: $S_n/a_n$ , a.s. convergence , expected value , stopping rule , supremum

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 5 • October, 1974
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