Open Access
May, 1985 Optimal Stopping of Independent Random Variables and Maximizing Prophets
D. P. Kennedy
Ann. Probab. 13(2): 566-571 (May, 1985). DOI: 10.1214/aop/1176993009


The prophet inequality for a sequence of independent nonnegative random variables shows that the ratio of the mean of the maximum of the sequence to the optimal expected return using stopping times is always bounded by 2; i.e., on average, the proportional advantage of a prophet with complete foresight over a gambler using nonanticipating stopping rules is at most 2. Here, an inequality linking the mean of the sum of the $k$ largest order statistics of the sequence and the optimal expected return is derived. This implies that if the $k$ largest order statistics are close to the maximum in mean then the proportional advantage of the prophet is at most of order $(k + 1)/k$. An extension of the additive prophet inequality for uniformly bounded independent random variables is also given.


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D. P. Kennedy. "Optimal Stopping of Independent Random Variables and Maximizing Prophets." Ann. Probab. 13 (2) 566 - 571, May, 1985.


Published: May, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0563.60044
MathSciNet: MR781423
Digital Object Identifier: 10.1214/aop/1176993009

Primary: 60G40
Secondary: 62L15

Keywords: Optimal stopping , order statistics , prophet inequality

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 2 • May, 1985
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