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Compare the ratio of symmetric polynomials of odds to one and stop

Part of: Stochastic processes

Published online by Cambridge University Press:  04 April 2017

Tomomi Matsui  and
Katsunori Ano
Tomomi Matsui*
Affiliation:
Tokyo Institute of Technology
Katsunori Ano*
Affiliation:
Shibaura Institute of Technology
*
* Postal address: Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8550, Japan. Email address: [email protected]
** Postal address: Department of Mathematical Sciences, Shibaura Institute of Technology, Fukasaku, Minuma-ku, Saitama-shi, Saitama, 337-8570, Japan.
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Abstract

In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last ℓ successes, given a sequence of independent Bernoulli trials of length N, where k and ℓ are predetermined integers satisfying 1≤k≤ℓ<N. This problem includes some odds problems as special cases, e.g. Bruss’ odds problem, Bruss and Paindaveine’s problem of selecting the last ℓ successes, and Tamaki’s multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton’s inequalities and optimization technique, which gives a unified view to the previous works.

Keywords

Optimal stoppingodds problemlower boundsecretary problemNewton’s inequality

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 12 - 22
Copyright
Copyright © Applied Probability Trust 2017 

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