## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 3 (1976), 418-432.

### The Infinite Secretary Problem

Jacqueline Gianini and Stephen M. Samuels

#### Abstract

An infinite sequence of rankable individuals (rank $1 =$ best) arrive at times which are i.i.d., uniform on (0, 1). We, in effect, observe only their relative ranks as they arrive. We seek a stopping rule to minimize the mean of a prescribed positive increasing function, $q(\bullet)$, of the actual rank of the individual chosen. Let $f(t)$ be the minimal mean among all stopping rules which are greater than $t$. Then $f(\bullet)$ is a solution to a certain differential equation which is derived and used to find an optimal stopping rule. This problem is in a strong sense the "limit" of a corresponding sequence of "finite secretary problems" which have been examined by various authors. The limit of the "finite-problem" minimal risks is finite if and only if the differential equation has a solution, $f(\bullet)$, which is finite on $\lbrack 0, 1)$ with $f(1^-) = \sup q(n)$. Usually, if such a solution exists, it is unique, in which case $f(0)$ is both the minimal risk for the infinite problem and the limit of the "finite-problem" minimal risks.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 3 (1976), 418-432.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996090

**Digital Object Identifier**

doi:10.1214/aop/1176996090

**Mathematical Reviews number (MathSciNet)**

MR413389

**Zentralblatt MATH identifier**

0341.60033

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Optimal stopping rules loss function relative ranks

#### Citation

Gianini, Jacqueline; Samuels, Stephen M. The Infinite Secretary Problem. Ann. Probab. 4 (1976), no. 3, 418--432. doi:10.1214/aop/1176996090. https://projecteuclid.org/euclid.aop/1176996090