Open Access
December, 1970 On the Supremum of $S_n/n$
B. J. McCabe, L. A. Shepp
Ann. Math. Statist. 41(6): 2166-2168 (December, 1970). DOI: 10.1214/aoms/1177696723


Let $X_1, X_2, \cdots$ be independent and identically distributed. We give a simple proof based on stopping times of the known result that $\sup(|X_1 + \cdots + X_n|/n)$ has a finite expected value if and only if $E|X| \log |X|$ is finite. Whenever $E|X| \log |X| = \infty$, a simple nonanticipating stopping rule $\tau$, not depending on $X$, yields $E(|X_1 + \cdots + X_\tau|/\tau) = \infty$.


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B. J. McCabe. L. A. Shepp. "On the Supremum of $S_n/n$." Ann. Math. Statist. 41 (6) 2166 - 2168, December, 1970.


Published: December, 1970
First available in Project Euclid: 27 April 2007

zbMATH: 0226.60067
MathSciNet: MR267627
Digital Object Identifier: 10.1214/aoms/1177696723

Rights: Copyright © 1970 Institute of Mathematical Statistics

Vol.41 • No. 6 • December, 1970
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