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Optimal Online Selection of an Alternating Subsequence: A Central Limit Theorem

Part of:Combinatorial probabilityStochastic processesLimit theorems

Published online by Cambridge University Press:  22 February 2016

Alessandro Arlotto and
J. Michael Steele
Alessandro Arlotto*
Affiliation:
Duke University
J. Michael Steele*
Affiliation:
University of Pennsylvania
*
∗ Postal address: The Fuqua School of Business, Duke University, 100 Fuqua Drive, Durham, NC 27708, USA. Email address: alessandro.arlotto@duke.edu
∗∗ Postal address: The Wharton School, University of Pennsylvania, 3730 Walnut Street, Philadelphia, PA 19104, USA. Email address: steele@wharton.upenn.edu
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Abstract

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We analyze the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution F, and we prove a central limit theorem for the number of selections made by that policy. The proof exploits the backward recursion of dynamic programming and assembles a detailed understanding of the associated value functions and selection rules.

Keywords

Bellman equationonline selectionMarkov decision problemdynamic programmingalternating subsequencecentral limit theoremnonhomogeneous Markov chain

MSC classification

Primary:60C05: Combinatorial probability60G40: Stopping times; optimal stopping problems; gambling theory90C40: Markov and semi-Markov decision processes
Secondary:60F05: Central limit and other weak theorems90C27: Combinatorial optimization90C39: Dynamic programming

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014, pp. 536 - 559
Copyright
© Applied Probability Trust 

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