{
  "id": "math/0508464",
  "version": "v1",
  "published": "2005-08-24T14:51:08.000Z",
  "updated": "2005-08-24T14:51:08.000Z",
  "title": "Convergence of random measures in geometric probability",
  "authors": [
    "Mathew D. Penrose"
  ],
  "comment": "51 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\\nu_n = \\sum_i \\xi_i $, where $\\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a law of large numbers and central limit theorem for $\\nu_n(f)$. The latter implies weak convergence of $\\nu_n(\\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications including the volume and surface measure of germ-grain models with unbounded grain sizes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-24T14:51:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60G57",
      "60F05",
      "60F25",
      "52A22"
    ],
    "keywords": [
      "geometric probability",
      "random measures",
      "bounded test functions",
      "central limit theorem",
      "implies weak convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8464P"
    }
  }
}