{
  "id": "1101.4887",
  "version": "v2",
  "published": "2011-01-25T17:55:22.000Z",
  "updated": "2013-10-21T05:51:24.000Z",
  "title": "A multivariate Gnedenko law of large numbers",
  "authors": [
    "Daniel Fresen"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/12-AOP804 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2013, Vol. 41, No. 5, 3051-3080",
  "doi": "10.1214/12-AOP804",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case. We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-21T05:51:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multivariate gnedenko law",
      "large numbers",
      "absolutely continuous log-concave distribution approximates",
      "logarithmic hausdorff distance",
      "convex hull"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.4887F"
    }
  }
}