## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 3 (1992), 1222-1228.

### A Difference Prophet Inequality for Bounded I.I.D. Variables, with Cost for Observations

#### Abstract

Let $X_i$ be i.i.d. random variables, $0 \leq X_i \leq 1$ and $c \geq 0$, and let $Y_i = X_i - ic$. It is shown that for all $n$, all $c$ and all such $X_i, E(\max_{i \geq 1} Y_i) - \sup_t EY_t < e^{-1}$, where $t$ is a stopping rule and $e^{-1}$ is shown to be the best bound for which the inequality holds. Specific bounds are also obtained for fixed $n$ or fixed $c$. These results are very similar to those obtained by Jones for a similar problem, where $0 \leq X_i \leq 1$ are independent but not necessarily identically distributed. All results are valid and unchanged also when $Y_i$ is replaced by $Y^\ast_i = \max_{1 \leq j \leq i} X_j - ic$.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 3 (1992), 1222-1228.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989689

**Digital Object Identifier**

doi:10.1214/aop/1176989689

**Mathematical Reviews number (MathSciNet)**

MR1175260

**Zentralblatt MATH identifier**

0777.60035

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60E15: Inequalities; stochastic orderings

**Keywords**

Prophet inequality optimal stopping cost of observation

#### Citation

Samuel-Cahn, Ester. A Difference Prophet Inequality for Bounded I.I.D. Variables, with Cost for Observations. Ann. Probab. 20 (1992), no. 3, 1222--1228. doi:10.1214/aop/1176989689. https://projecteuclid.org/euclid.aop/1176989689