Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T21:36:51.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone Case for an Extended Process

Part of: Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Małgorzata Kuchta  and
Michał Morayne
Małgorzata Kuchta*
Affiliation:
Wrocław University of Technology
Michał Morayne*
Affiliation:
Wrocław University of Technology
*
∗ Postal address: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland.
∗ Postal address: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a nonnegative discrete time and bounded horizon process X for which 0 is an absorbing state and extend it by a random variable that is independent of X. We find a sufficient condition for the resulting process to satisfy, after a canonical time rescaling, the hypothesis of the monotone case theorem. If X describes a secretary type search on a poset with one maximal element or if we consider X with no extension then this condition assumes an especially simple log-concavity type form.

Keywords

Monotone casesecretary problembest choiceoptimal stopping time

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1106 - 1125
Copyright
© Applied Probability Trust 

Footnotes

This research was partially supported by the MNiSW grant NN 206 36 9739.

References

Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.Google Scholar
Freij, R. and Wästlund, J. (2010). Partially ordered secretaries. Electron. Commun. Prob. 15, 504–507.Google Scholar
Garrod, B. and Morris, R. (2013). The secretary problem on an unknown poset. Random Structures Algorithms 43, 429–451.Google Scholar
Garrod, B., Kubicki, G. and Morayne, M. (2012). How to choose the best twins. SIAM J. Discrete Math. 26, 384–398.Google Scholar
Georgiou, N., Kuchta, M., Morayne, M. and Niemiec, J. (2008). On a universal best choice algorithm for partially ordered sets. Random Structures Algorithms 32, 263–273.Google Scholar
Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.CrossRefGoogle Scholar
Kaźmierczak, W. (2013). The best choice problem for a union of two linear orders with common maximum. Discrete Appl. Math. 161, 3090–3096.Google Scholar
Kozik, J. (2010). Dynamic threshold strategy for universal best choice problem. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 439–451.Google Scholar
Kuchta, M. (2014). Iterated full information secretary problem. Submitted.Google Scholar
Kuchta, M. and Morayne, M. (2014). A secretary problem with many lives. Commun. Statist. Theory Meth. 43, 210–218.Google Scholar
Kumar, R., Lattanzi, S., Vassilvitskii, S. and Vattani, A. (2011). Hiring a secretary from a poset. In Proc. ACM Conf. Electron. Commerce, ACM, New York, pp. 39–48.Google Scholar
Kurpisz, A. and Morayne, M. (2014). Inform friends, do not inform enemies. IMA J. Math. Control Inf. 31, 435–440.Google Scholar
Morayne, M. (1998). Partial-order analogue of the secretary problem: the binary tree case. Discrete Math. 184, 165–181.CrossRefGoogle Scholar
Preater, J. (1999). The best-choice problem for partially ordered objects. Operat. Res. Lett. 25, 187–190.Google Scholar
Samuels, S. M. (1991). Secretary problems. In Handbook of Sequential Analysis, Dekker, New York, pp. 381–405.Google Scholar