{
  "id": "0906.4238",
  "version": "v1",
  "published": "2009-06-23T12:21:10.000Z",
  "updated": "2009-06-23T12:21:10.000Z",
  "title": "Poisson--Voronoi approximation",
  "authors": [
    "Matthias Heveling",
    "Matthias Reitzner"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AAP561 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2009, Vol. 19, No. 2, 719-736",
  "doi": "10.1214/08-AAP561",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a Poisson point process and $K\\subset\\mathbb{R}^d$ a measurable set. Construct the Voronoi cells of all points $x\\in X$ with respect to $X$, and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in $K$. For $K$ a compact convex set the expectation of the volume difference $V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\\triangle K)$ is computed. Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a Poisson point process. Concentration inequalities for both quantities are proved using Azuma's inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-23T12:21:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60G55",
      "52A22",
      "60C05"
    ],
    "keywords": [
      "poisson-voronoi approximation",
      "poisson point process",
      "voronoi cells",
      "compact convex set",
      "volume difference"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.4238H"
    }
  }
}