## Journal of Applied Probability

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

#### 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 X1,...,Xn are independent random variables with known continuous distributions and Vn(X1,...,Xn):=supτℙ(Xτ=Mn), where Mn≔max{X1,...,Xn} and the supremum is over all stopping times adapted to X1,...,Xn then Vn(X1,...,Xn)≥(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

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

Mathematical Reviews number (MathSciNet)
MR3581240

Zentralblatt MATH identifier
1355.60055

#### 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