## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 2 (1983), 442-450.

### The Advantage of Using Non-Measurable Stop Rules

Theodore P. Hill and Victor C. Pestien

#### Abstract

Comparisons are made between the expected returns using measurable and non-measurable stop rules in discrete-time stopping problems. In the independent case, a natural sufficient condition ("preservation of independence") is found for the expected return of every bounded non-measurable stopping function to be equal to that of a measurable one, and for that of every unbounded non-measurable stopping function to be arbitrarily close to that of a measurable one. For non-negative and for uniformly-bounded independent random variables, universal sharp bounds are found for the advantage of using non-measurable stopping functions over using measurable ones. Partial results for the dependent case are obtained.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 2 (1983), 442-450.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993609

**Digital Object Identifier**

doi:10.1214/aop/1176993609

**Mathematical Reviews number (MathSciNet)**

MR690141

**Zentralblatt MATH identifier**

0529.60041

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 90C39: Dynamic programming [See also 49L20]

**Keywords**

Optimal stopping theory non-measurable stopping function stop rule

#### Citation

Hill, Theodore P.; Pestien, Victor C. The Advantage of Using Non-Measurable Stop Rules. Ann. Probab. 11 (1983), no. 2, 442--450. doi:10.1214/aop/1176993609. https://projecteuclid.org/euclid.aop/1176993609