Journal of Applied Probability

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

Pieter C. Allaart and José A. Islas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

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


Export citation

References