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

Part of: Stochastic processes Combinatorial probability Limit 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: [email protected]
∗∗ Postal address: The Wharton School, University of Pennsylvania, 3730 Walnut Street, Philadelphia, PA 19104, USA. Email address: [email protected]
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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 probability 60G40: Stopping times; optimal stopping problems; gambling theory 90C40: Markov and semi-Markov decision processes
Secondary: 60F05: Central limit and other weak theorems 90C27: Combinatorial optimization 90C39: Dynamic programming
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 536 - 559
Copyright
© Applied Probability Trust 

References

Arlotto, A. and Steele, J. M. (2011). Optimal sequential selection of a unimodal subsequence of a random sequence. Combin. Prob. Comput. 20, 799–814.Google Scholar
Arlotto, A., Chen, R. W., Shepp, L. A. and Steele, J. M. (2011). Online selection of alternating subsequences from a random sample. J. Appl. Prob. 48, 1114–1132.Google Scholar
Bannister, M. J. and Eppstein, D. (2012). Randomized speedup of the Bellman–Ford algorithm. In Proc. Meeting on Analytic Algorithmics and Combinatorics. Society for Industrial and Applied Mathematics, Philadelphia, PA, pp. 41–47.Google Scholar
Baryshnikov, Y. M. and Gnedin, A. V. (2000). Sequential selection of an increasing sequence from a multidimensional random sample. Ann. Appl. Prob. 10, 258–267.CrossRefGoogle Scholar
Bateni, M., Hajiaghayi, M. and Zadimoghaddam, M. (2010). Submodular secretary problem and extensions. In Approximation, Randomization, and Combinatorial Optimization (Lecture Notes Comput. Sci. 6302), Springer, Berlin, pp. 39–52.Google Scholar
Brockwell, P. J. and Davis, R. A. (2006). Time Series: Theory and Methods. Springer, New York.Google Scholar
Bruss, F. T. and Delbaen, F. (2001). Optimal rules for the sequential selection of monotone subsequences of maximum expected length. Stoch. Process. Appl. 96, 313–342.Google Scholar
Bruss, F. T. and Delbaen, F. (2004). A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length. Stoch. Process. Appl. 114, 287–311.CrossRefGoogle Scholar
Buchbinder, N., Jain, K. and Singh, M. (2010). Secretary problems via linear programming. In Integer Programming and Combinatorial Optimization (Lecture Notes Comput. Sci. 6080), Springer, Berlin, pp. 163–176.Google Scholar
Cayley, A. (1875). Mathematical questions and their solutions. Educational Times 22, 18–19. (Available in The Collected Mathematical Papers of Arthur Cayley (1986), Vol. 10, Cambridge University Press, pp. 587-588.)Google Scholar
Dynkin, E. B. (1963). Optimal choice of the stopping moment of a Markov process. Dokl. Akad. Nauk SSSR 150, 238–240.Google Scholar
Gnedin, A. V. (1999). Sequential selection of an increasing subsequence from a sample of random size. J. Appl. Prob. 36, 1074–1085.CrossRefGoogle Scholar
Gnedin, A. V. (2000a). A note on sequential selection from permutations. Combin. Prob. Comput. 9, 13–17.Google Scholar
Gnedin, A. V. (2000b). Sequential selection of an increasing subsequence from a random sample with geometrically distributed sample-size. In Game Theory, Optimal Stopping, Probability and Statistics (IMS Lecture Notes Monogr. Ser. 35), Institute of Mathematical Statistics, Beachwood, OH, pp. 101–109.Google Scholar
Houdré, C., and Restrepo, R. (2010). A probabilistic approach to the asymptotics of the length of the longest alternating subsequence. Electron. J. Combin. 17, Research Paper 168, 19pp.Google Scholar
Hua, X. (2010). Testing regression models with residuals as data. , Massachusetts Institute of Technology.Google Scholar
Jones, G. L. (2004). On the Markov chain central limit theorem. Prob. Surv. 1, 299–320.Google Scholar
Krieger, A. M. and Samuel-Cahn, E. (2009). The secretary problem of minimizing the expected rank: a simple suboptimal approach with generalizations. Adv. Appl. Prob. 41, 1041–1058.Google Scholar
Lindley, D. V. (1961). Dynamic programming and decision theory. Appl. Statist. 10, 39–51.Google Scholar
Lindvall, T. (2002). Lectures on the Coupling Method. Dover, Mineola, NY.Google Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
Romik, D. (2011). Local extrema in random permutations and the structure of longest alternating subsequences. In 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 825–834.Google Scholar
Samuels, S. M. (1991). Secretary problems. In Handbook of Sequential Analysis (Statist. Textbooks Monogr. 118), Dekker, New York, pp. 381–405.Google Scholar
Samuels, S. M. and Steele, J. M. (1981). Optimal sequential selection of a monotone sequence from a random sample. Ann. Prob. 9, 937–947.Google Scholar
Stanley, R. P. (2007). Increasing and decreasing subsequences and their variants. In International Congress of Mathematicians, Vol. I, European Mathematical Society, Zürich, pp. 545–579.Google Scholar
Stanley, R. P. (2008). Longest alternating subsequences of permutations. Michigan Math. J. 57, 675–687.Google Scholar
Stanley, R. P. (2010). A survey of alternating permutations. In Combinatorics and Graphs (Contemp. Math. 531), American Mathematical Society, Providence, RI, pp. 165–196.Google Scholar
Widom, H. (2006). On the limiting distribution for the length of the longest alternating sequence in a random permutation. Electron. J. Combin. 13, Research Paper 25, 7pp.Google Scholar