## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 2 (1993), 898-916.

### The Infinite Secretary Problem with Recall

#### Abstract

The infinite secretary problem, in which an infinite number of rankable items arrive at times which are i.i.d., uniform on (0, 1), is modified to allow for a fixed period of recall of length $\alpha, 0 \leq \alpha \leq 1$. The goal is to find the maximum probability of best choice, $v = v(\alpha)$, as well as an optimal stopping time $\tau^\ast = \tau^\ast(\alpha)$. A differential-delay equation is derived, the solution of which yields $v(\alpha)$ and $\tau^\ast(\alpha)$, the latter given in terms of a constant $t^\ast \lbrack = t^\ast(\alpha)\rbrack$. For $\alpha \geq 1/2$, the complete solution to the problem is obtained. For $0 < \alpha < 1/2, v(\alpha)$ cannot be put in closed form, so upper and lower bounds for $v(\alpha)$ and $t^\ast(\alpha)$ are obtained and are investigated for $\alpha$ near 0 and near 1/2, where the solutions are known. We also find asymptotic expansions of $v(\alpha)$ and $t^\ast(\alpha)$ about $\alpha = 0$ and $\alpha = 1/2$. Finally, the solution to the finite, $n$-item length-$m$ recall problem introduced by Smith and Deely is shown to converge to the solution of the infinite problem when $m/n \rightarrow \alpha$.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 2 (1993), 898-916.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989273

**Digital Object Identifier**

doi:10.1214/aop/1176989273

**Mathematical Reviews number (MathSciNet)**

MR1217571

**Zentralblatt MATH identifier**

0776.60057

**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]

**Keywords**

Secretary problem optimal stopping secretary problem with recall best selection best choice problem asymptotic analysis

#### Citation

Rocha, Amy L. The Infinite Secretary Problem with Recall. Ann. Probab. 21 (1993), no. 2, 898--916. doi:10.1214/aop/1176989273. https://projecteuclid.org/euclid.aop/1176989273