Maximizing $E \max_{1 \leq k \leq n} S^+_k/ES^+_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates
Klass, Michael J.
Ann. Probab., Tome 17 (1989) no. 4, p. 1243-1247 / Harvested from Project Euclid
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Let $X, X_1, X_2, \ldots$ be i.i.d. mean zero random variables. Put $S_k = X_1 + \cdots + X_k$. We prove that for every $n \geq 1, E \max_{1 \leq k \leq n} S^+_n \leq (2 - n^{-1})ES^+_n$. This result is nearly sharp, since if $P(X = 1) = P(X = -1) = \frac{1}{2},$ then $E \max{1 \leq k \leq n} S^+_k = (2 - n^{-1/2}\gamma^+_n)ES^+_n,$ where $\lim_{n \rightarrow \infty} \gamma^+_n = \sqrt{\pi/2}$.
Publié le : 1989-07-14
Classification:  Maximum of partial sums,  prophet inequalities,  60E15,  60G50,  60G40,  60J15 
@article{1176991266,
     author = {Klass, Michael J.},
     title = {Maximizing $E \max\_{1 \leq k \leq n} S^+\_k/ES^+\_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates},
     journal = {Ann. Probab.},
     volume = {17},
     number = {4},
     year = {1989},
     pages = { 1243-1247},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176991266}
}

Klass, Michael J. Maximizing $E \max_{1 \leq k \leq n} S^+_k/ES^+_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates. Ann. Probab., Tome 17 (1989) no. 4, pp.  1243-1247. http://gdmltest.u-ga.fr/item/1176991266/