Diffusion approximations in the online increasing subsequence problem

Alexander Gnedin and Amirlan Seksenbayev

Stochastic Processes and their Applications, 2021, vol. 139, issue C, 298-320

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.

Keywords: Increasing subsequence; Online selection strategy; Functional limits (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.spa.2021.06.001

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