## Journal of Applied Probability

- J. Appl. Probab.
- Volume 53, Number 4 (2016), 1041-1051.

### A sharp lower bound for choosing the maximum of an independent sequence

Pieter C. Allaart and José A. Islas

#### Abstract

In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if *X*_{1},...,*X*_{n} are independent random variables with known continuous distributions and *V*_{n}(*X*_{1},...,X_{n}):=sup_{τ}ℙ(*X*_{τ}=*M*_{n}), where *M*_{n}≔max{*X*_{1},...,*X*_{n}} and the supremum is over all stopping times adapted to *X*_{1},...,*X*_{n} then *V*_{n}(*X*_{1},...,*X*_{n})≥(1-1/*n*)^{n-1}, and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds theorem, Bruss (2000). In order to obtain a sharp bound for each *n*, we improve Bruss' lower bound, Bruss (2003), for the sum-the-odds problem.

#### Article information

**Source**

J. Appl. Probab., Volume 53, Number 4 (2016), 1041-1051.

**Dates**

First available in Project Euclid: 7 December 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1481132835

**Mathematical Reviews number (MathSciNet)**

MR3581240

**Zentralblatt MATH identifier**

1355.60055

**Subjects**

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

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

**Keywords**

Choosing the maximum sum-the-odds theorem stopping time

#### Citation

Allaart, Pieter C.; Islas, José A. A sharp lower bound for choosing the maximum of an independent sequence. J. Appl. Probab. 53 (2016), no. 4, 1041--1051. https://projecteuclid.org/euclid.jap/1481132835