The Annals of Probability

Maximizing $E \max_{1 \leq k \leq n} S^+_k/ES^+_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates

Michael J. Klass

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Abstract

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 $\lim_{n \rightarrow \infty} \gamma^+_n = \sqrt{\pi/2}$.

Article information

Source
Ann. Probab., Volume 17, Number 3 (1989), 1243-1247.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991266

Digital Object Identifier
doi:10.1214/aop/1176991266

Mathematical Reviews number (MathSciNet)
MR1009454

Zentralblatt MATH identifier
0684.60032

JSTOR
links.jstor.org

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G50: Sums of independent random variables; random walks 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J15

Keywords
Maximum of partial sums prophet inequalities

Citation

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. 17 (1989), no. 3, 1243--1247. doi:10.1214/aop/1176991266. https://projecteuclid.org/euclid.aop/1176991266


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