Date of this Version
The Annals of Mathematical Statistics
Let X1,X2,⋯ be independent and identically distributed. We give a simple proof based on stopping times of the known result that sup ( | X1 + ⋯ + Xn|/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) = ∞.
McCabe, B. J., & Shepp, L. A. (1970). On the Supremum of Sn/n. The Annals of Mathematical Statistics, 41 (6), 2166-2168. http://dx.doi.org/10.1214/aoms/1177696723
Date Posted: 27 November 2017