## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 2 (1982), 336-345.

### Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables

T. P. Hill and Robert P. Kertz

#### Abstract

Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if $X_1, X_2, \cdots$ are i.i.d. non-negative random variables and if $T_n$ is the set of stop rules for $X_1, \cdots, X_n$, then $E(\max\{X_1, \cdots, X_n\}) \leq a_n \sup\{EX_t: t \in T_n\}$, and the bound $a_n$ is best possible. Similar universal constants $0 < b_n < \frac{1}{4}$ are found so that if the $\{X_i\}$ are i.i.d. random variables taking values only in $\lbrack a, b\rbrack$, then $E(\max\{X_1, \cdots, X_n\}) \leq \sup\{EX_t: t \in T_n\} + b_n(b - a)$, where again the bound $b_n$ is best possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 2 (1982), 336-345.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993861

**Digital Object Identifier**

doi:10.1214/aop/1176993861

**Mathematical Reviews number (MathSciNet)**

MR647508

**Zentralblatt MATH identifier**

0483.60035

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62L15: Optimal stopping [See also 60G40, 91A60] 90C99: None of the above, but in this section

**Keywords**

Optimal stopping extremal distributions inequalities for stochastic processes

#### Citation

Hill, T. P.; Kertz, Robert P. Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables. Ann. Probab. 10 (1982), no. 2, 336--345. doi:10.1214/aop/1176993861. https://projecteuclid.org/euclid.aop/1176993861