{
  "id": "0909.3642",
  "version": "v2",
  "published": "2009-09-20T16:44:12.000Z",
  "updated": "2009-11-20T08:22:26.000Z",
  "title": "Characterizations of exchangeable partitions and random discrete distributions by deletion properties",
  "authors": [
    "Alexander Gnedin",
    "Chris Haulk",
    "Jim Pitman"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove a long-standing conjecture which characterises the Ewens-Pitman two-parameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each $n = 2,3, >...$, if one of $n$ individuals is chosen uniformly at random, independently of the random partition $\\pi_n$ of these individuals into various types, and all individuals of the same type as the chosen individual are deleted, then for each $r > 0$, given that $r$ individuals remain, these individuals are partitioned according to $\\pi_r'$ for some sequence of random partitions $(\\pi_r')$ that does not depend on $n$. An analogous result characterizes the associated Poisson-Dirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a size-biased pick. We also survey the regenerative properties of members of the two-parameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the Poisson-Dirichlet random sequence into the interval partition induced by the range of a neutral-to-the right process.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-20T08:22:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G09"
    ],
    "keywords": [
      "random discrete distributions",
      "deletion properties",
      "exchangeable partitions",
      "individual",
      "characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.3642G"
    }
  }
}