## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 3 (1989), 1243-1247.

### Maximizing $E \max_{1 \leq k \leq n} S^+_k/ES^+_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates

#### Abstract

Let $X, X_1, X_2, \ldots$ be i.i.d. mean zero random variables. Put $S_k = X_1 + \cdots + X_k$. We prove that for every $n \geq 1, E \max_{1 \leq k \leq n} S^+_n \leq (2 - n^{-1})ES^+_n$. This result is nearly sharp, since if $P(X = 1) = P(X = -1) = \frac{1}{2},$ then $E \max{1 \leq k \leq n} S^+_k = (2 - n^{-1/2}\gamma^+_n)ES^+_n,$ where $\lim_{n \rightarrow \infty} \gamma^+_n = \sqrt{\pi/2}$.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 3 (1989), 1243-1247.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991266

**Digital Object Identifier**

doi:10.1214/aop/1176991266

**Mathematical Reviews number (MathSciNet)**

MR1009454

**Zentralblatt MATH identifier**

0684.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E15: Inequalities; stochastic orderings

Secondary: 60G50: Sums of independent random variables; random walks 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J15

**Keywords**

Maximum of partial sums prophet inequalities

#### Citation

Klass, Michael J. Maximizing $E \max_{1 \leq k \leq n} S^+_k/ES^+_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates. Ann. Probab. 17 (1989), no. 3, 1243--1247. doi:10.1214/aop/1176991266. https://projecteuclid.org/euclid.aop/1176991266