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Diffusion approximations in the online increasing subsequence problem

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  • Gnedin, Alexander
  • Seksenbayev, Amirlan

Abstract

The online increasing subsequence problem is a stochastic optimisation task with the objective to maximise the expected length of subsequence chosen from a random series by means of a nonanticipating decision strategy. We study the structure of optimal and near-optimal subsequences in a standardised planar Poisson framework. Following a long-standing suggestion by Bruss and Delbaen (2004), we prove a joint functional limit theorem for the transversal fluctuations about the diagonal of the running maximum and the length processes. The limit is identified explicitly with a Gaussian time-inhomogeneous diffusion. In particular, the running maximum converges to a Brownian bridge, and the length process has another explicit non-Markovian limit.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:139:y:2021:i:c:p:298-320
    DOI: 10.1016/j.spa.2021.06.001
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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. 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.
    3. 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.
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