## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 4 (1987), 1593-1599.

### Prophet Compared to Gambler: An Inequality for Transforms of Processes

Ulrich Krengel and Louis Sucheston

#### Abstract

A prophet is a player with complete foresight; a gambler knows only the past and the present, but not the future. If each of them bets on differences of consecutive nonnegative random variables $X_i$ such that $E(X_i|X_{i - 1}) = EX_i$, the players multiplying their stakes by uniformly bounded variables, then the expected gain of the prophet is at most three times that of the gambler. The constant 3 is optimal.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 4 (1987), 1593-1599.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991996

**Digital Object Identifier**

doi:10.1214/aop/1176991996

**Mathematical Reviews number (MathSciNet)**

MR905351

**Zentralblatt MATH identifier**

0637.60057

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Prophet inequality transforms of processes

#### Citation

Krengel, Ulrich; Sucheston, Louis. Prophet Compared to Gambler: An Inequality for Transforms of Processes. Ann. Probab. 15 (1987), no. 4, 1593--1599. doi:10.1214/aop/1176991996. https://projecteuclid.org/euclid.aop/1176991996