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On the Supremum of Sn/n
B. J. McCabe and L. A. Shepp
The Annals of Mathematical Statistics
Vol. 41, No. 6 (Dec., 1970), pp. 2166-2168
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2240360
Page Count: 3
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Let X1, X2, ⋯ 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| = ∞, a simple nonanticipating stopping rule τ, not depending on X, yields E(|X1 + ⋯ + Xτ|/τ) = ∞.
The Annals of Mathematical Statistics © 1970 Institute of Mathematical Statistics