## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 6 (1981), 937-947.

### Optimal Sequential Selection of a Monotone Sequence From a Random Sample

Stephen M. Samuels and J. Michael Steele

#### Abstract

The length of the longest monotone increasing subsequence of a random sample of size $n$ is known to have expected value asymptotic to $2n^{1/2}$. We prove that it is possible to make sequential choices which give an increasing subsequence of expected length asymptotic to $(2n)^{1/2}$. Moreover, this rate of increase is proved to be asymptotically best possible.

#### Article information

**Source**

Ann. Probab. Volume 9, Number 6 (1981), 937-947.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176994265

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176994265

**Mathematical Reviews number (MathSciNet)**

MR632967

**Zentralblatt MATH identifier**

0473.62073

**Subjects**

Primary: 62L15: Optimal stopping [See also 60G40, 91A60]

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

**Keywords**

Monotone subsequence optimal stopping subadditive process

#### Citation

Samuels, Stephen M.; Steele, J. Michael. Optimal Sequential Selection of a Monotone Sequence From a Random Sample. The Annals of Probability 9 (1981), no. 6, 937--947. doi:10.1214/aop/1176994265. http://projecteuclid.org/euclid.aop/1176994265.