{
  "id": "2101.06306",
  "version": "v1",
  "published": "2021-01-15T21:51:23.000Z",
  "updated": "2021-01-15T21:51:23.000Z",
  "title": "Random Euclidean coverage from within",
  "authors": [
    "Mathew D. Penrose"
  ],
  "comment": "70 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_1,X_2, \\ldots $ be i.i.d. random uniform points in a bounded domain $A \\subset {\\mathbb{R}}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$ centred on $X_1,\\ldots,X_n$. We obtain the limiting distribution of $R_n$ and also a strong law of large numbers for $R_n$ in the large-$n$ limit. For example, if $d=3$ and $A$ has volume 1 and perimeter $|\\partial A|$ then $\\Pr[n\\pi R_n^3 - \\log n - 2 \\log (\\log n) \\leq x]$ converges to $\\exp(-2^{-4}\\pi^{5/3} |\\partial A| e^{-2 x/3})$, and $(n \\pi R_n^3)/(\\log n) \\to 1$ almost surely. We give similar results for general $d$, and also for the case where $A$ is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-15T21:51:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60F05",
      "60F15"
    ],
    "keywords": [
      "random euclidean coverage",
      "large numbers",
      "strong law",
      "random uniform points",
      "boundary effects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}