On the Supremum of $S_n/n$
McCabe, B. J. ; Shepp, L. A.
Ann. Math. Statist., Tome 41 (1970) no. 6, p. 2166-2168 / Harvested from Project Euclid
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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 $E(|X_1 + \cdots + X_\tau|/\tau) = \infty$.
Publié le : 1970-12-14
Classification:  
@article{1177696723,
     author = {McCabe, B. J. and Shepp, L. A.},
     title = {On the Supremum of $S\_n/n$},
     journal = {Ann. Math. Statist.},
     volume = {41},
     number = {6},
     year = {1970},
     pages = { 2166-2168},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177696723}
}

McCabe, B. J.; Shepp, L. A. On the Supremum of $S_n/n$. Ann. Math. Statist., Tome 41 (1970) no. 6, pp.  2166-2168. http://gdmltest.u-ga.fr/item/1177696723/