## The Annals of Probability

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

#### 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

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

#### 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