{
  "id": "2001.02249",
  "version": "v1",
  "published": "2020-01-07T19:05:01.000Z",
  "updated": "2020-01-07T19:05:01.000Z",
  "title": "Diffusion Approximations in the Online Increasing Subsequence Problem",
  "authors": [
    "Alexander Gnedin",
    "Amirlan Seksenbayev"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "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 (Stoch. Proc. Appl. 114, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-07T19:05:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "online increasing subsequence problem",
      "diffusion approximations",
      "length process",
      "joint functional limit theorem",
      "stochastic optimisation task"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}