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Selecting a sequence of last successes in independent trials

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss  and
Davy Paindaveine
F. Thomas Bruss*
Affiliation:
Université Libre de Bruxelles
Davy Paindaveine*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Département de Mathématique et ISRO, Université Libre de Bruxelles, Campus Plaine, CP 210, B-1050 Bruxelles, Belgium
∗Postal address: Département de Mathématique et ISRO, Université Libre de Bruxelles, Campus Plaine, CP 210, B-1050 Bruxelles, Belgium
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Abstract

Let I1,I2,…,In be a sequence of independent indicator functions defined on a probability space (Ω, A, P). We say that index k is a success time if Ik = 1. The sequence I1,I2,…,In is observed sequentially. The objective of this article is to predict the lth last success, if any, with maximum probability at the time of its occurrence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We then extend the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.

Keywords

Sum the odds algorithmoptimal stoppingmultiple stoppingstopping islandsgenerating functionsmodified secretary problemsunimodality

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 389 - 399
Copyright
Copyright © by the Applied Probability Trust 2000 

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