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Optimal online selection of a monotone subsequence: a central limit theorem

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  • Arlotto, Alessandro
  • Nguyen, Vinh V.
  • Steele, J. Michael

Abstract

Consider a sequence of n independent random variables with a common continuous distribution F, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy πn∗ that is optimal in the sense that it maximizes the expected value of Ln(πn∗), the number of selected observations. We investigate the distribution of Ln(πn∗); in particular, we obtain a central limit theorem for Ln(πn∗) and a detailed understanding of its mean and variance for large n. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality N where N is a Poisson random variable that is independent of the sequence.

Suggested Citation

  • Arlotto, Alessandro & Nguyen, Vinh V. & Steele, J. Michael, 2015. "Optimal online selection of a monotone subsequence: a central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3596-3622.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:9:p:3596-3622
    DOI: 10.1016/j.spa.2015.03.009
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    References listed on IDEAS

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    1. Bruss, F. Thomas & Delbaen, Freddy, 2004. "A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 287-311, December.
    2. Alessandro Arlotto & Noah Gans & J. Michael Steele, 2014. "Markov Decision Problems Where Means Bound Variances," Operations Research, INFORMS, vol. 62(4), pages 864-875, August.
    3. Bruss, F. Thomas & Delbaen, Freddy, 2001. "Optimal rules for the sequential selection of monotone subsequences of maximum expected length," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 313-342, December.
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    Cited by:

    1. Gnedin, Alexander & Seksenbayev, Amirlan, 2021. "Diffusion approximations in the online increasing subsequence problem," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 298-320.

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