## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 2 (1985), 566-571.

### Optimal Stopping of Independent Random Variables and Maximizing Prophets

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 2 (1985), 566-571.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993009

**Digital Object Identifier**

doi:10.1214/aop/1176993009

**Mathematical Reviews number (MathSciNet)**

MR781423

**Zentralblatt MATH identifier**

0563.60044

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 62L15: Optimal stopping [See also 60G40, 91A60]

**Keywords**

Optimal stopping prophet inequality order statistics

#### Citation

Kennedy, D. P. Optimal Stopping of Independent Random Variables and Maximizing Prophets. Ann. Probab. 13 (1985), no. 2, 566--571. doi:10.1214/aop/1176993009. https://projecteuclid.org/euclid.aop/1176993009