## Journal of Applied Probability

- J. Appl. Probab.
- Volume 51, Number 3 (2014), 885-889.

### A note on a lower bound for the multiplicative odds theorem of optimal stopping

Tomomi Matsui and Katsunori Ano

#### Abstract

In this note we present a bound of the optimal maximum probability for the
multiplicative odds theorem of optimal stopping theory. We deal with an optimal
stopping problem that maximizes the probability of stopping on any of the last
*m* successes of a sequence of independent Bernoulli trials of length
*N*, where *m* and *N* are predetermined integers satisfying
1 ≤ *m* < *N*. This problem is an extension of Bruss'
(2000) odds problem. In a previous work, Tamaki (2010) derived an optimal
stopping rule. We present a lower bound of the optimal probability.
Interestingly, our lower bound is attained using a variation of the well-known
secretary problem, which is a special case of the odds problem.

#### Article information

**Source**

J. Appl. Probab., Volume 51, Number 3 (2014), 885-889.

**Dates**

First available in Project Euclid: 5 September 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1409932681

**Digital Object Identifier**

doi:10.1239/jap/1409932681

**Mathematical Reviews number (MathSciNet)**

MR3256234

**Zentralblatt MATH identifier**

1312.60047

**Subjects**

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

Secondary: 60L15

**Keywords**

Optimal stopping odd problem lower bound secretary problem Maclaurin's inequality

#### Citation

Matsui, Tomomi; Ano, Katsunori. A note on a lower bound for the multiplicative odds theorem of optimal stopping. J. Appl. Probab. 51 (2014), no. 3, 885--889. doi:10.1239/jap/1409932681. https://projecteuclid.org/euclid.jap/1409932681