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Optimal stopping on patterns in strings generated by independent random variables

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss  and
Guy Louchard
F. Thomas Bruss*
Affiliation:
Université Libre de Bruxelles
Guy Louchard*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Université Libre de Bruxelles, Département de Mathématiques et ISRO, CP 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. Email address: tbruss@ulb.ac.be
∗∗ Postal address: Université Libre de Bruxelles, Département d’Informatique, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
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Abstract

Strings are generated by sequences of independent draws from a given alphabet. For a given pattern H of length l and an integer n ≥ l, our goal is to maximize the probability of stopping on the last appearance of the pattern H in a string of length n (if any), given that, if we choose to stop on an occurrence of H, we must do so right away. This contrasts with the goals of several other investigations on patterns in strings such as computing the expected occurrence time and the probability of finding exactly r patterns in a string of fixed length. Several motivations are given for our problem ranging from relatively simple best choice problems to more difficult stopping problems allowing for a variety of interesting applications. We solve this problem completely in the homogeneous case for uncorrelated patterns. However, several of these results extend immediately to the inhomogeneous case. In the homogeneous case, optimal success probabilities are shown to vary, depending on characteristics of the pattern, essentially between the value 1/e and a new asymptotic constant 0.619…. These results demonstrate a considerable difference between the true optimal success probability compared with what a first approximation by heuristic arguments would yield. It is interesting to see that the so-called odds algorithm which gives the optimal rule for l = 1 yields an excellent approximation of the optimal rule for any l. This is important for applications because the odds algorithm is very convenient. We finally give a detailed asymptotic analysis for the homogeneous case.

Keywords

Odds algorithmsimple odds algorithmalgorithms on wordsinvestment problemssecretary problemsapartment problemsingle-island stopping ruleonline decisionspattern correlationPoisson clumping heuristicasymptotic analysis

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 49 - 72
Copyright
Copyright © Applied Probability Trust 2003 

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