{
  "id": "0911.5377",
  "version": "v3",
  "published": "2009-11-28T05:42:58.000Z",
  "updated": "2013-03-16T21:51:58.000Z",
  "title": "Poisson Thickening",
  "authors": [
    "Ori Gurel-Gurevich",
    "Ron Peled"
  ],
  "comment": "Added conjecture about when a deterministic coupling satisfying a relation exists. Made some minor revisions. To appear in Israel Journal of Mathematics. 16 pages",
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "Let X be a Poisson point process of intensity lambda on the real line. A thickening of it is a (deterministic) measurable function f such that the union of X and f(X) is a Poisson point process of intensity lambda' where lambda'>lambda. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo. We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-03-16T21:51:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50"
    ],
    "keywords": [
      "poisson point process",
      "poisson thickening",
      "intensity lambda",
      "real line",
      "deterministic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.5377G"
    }
  }
}