## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 4 (1984), 1213-1216.

### Comparison of Threshold Stop Rules and Maximum for Independent Nonnegative Random Variables

#### Abstract

Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 4 (1984), 1213-1216.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993150

**Digital Object Identifier**

doi:10.1214/aop/1176993150

**Mathematical Reviews number (MathSciNet)**

MR757778

**Zentralblatt MATH identifier**

0549.60036

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Stopping rules threshold rules prophet inequalities

#### Citation

Samuel-Cahn, Ester. Comparison of Threshold Stop Rules and Maximum for Independent Nonnegative Random Variables. Ann. Probab. 12 (1984), no. 4, 1213--1216. doi:10.1214/aop/1176993150. https://projecteuclid.org/euclid.aop/1176993150