{
  "id": "2401.03832",
  "version": "v1",
  "published": "2024-01-08T11:42:52.000Z",
  "updated": "2024-01-08T11:42:52.000Z",
  "title": "Covering one point process with another",
  "authors": [
    "Frankie Higgs",
    "Mathew D. Penrose",
    "Xiaochuan Yang"
  ],
  "comment": "39 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_1,X_2, \\ldots $ and $Y_1, Y_2, \\ldots$ be i.i.d. random uniform points in a bounded domain $A \\subset \\mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \\in \\mathbb{N}$, define the {\\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \\{Y_1,\\ldots,Y_m\\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \\to \\infty$ with $m= m(n) \\sim \\tau n$ for some constant $\\tau >0$, with $k $ fixed. If $A$ has unit area, then $n \\pi R_{n,m(n),1}^2 - \\log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\\log \\tau$. For $k >2$, we find that $n \\pi R_{n,m(n),k}^2 - \\log n - (2k-3) \\log \\log n$ is asymptotically Gumbel with scale parameter $2$ and a more complicated location parameter involving the perimeter of $A$; boundary effects dominate when $k >2$. For $k=2$ the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-08T11:42:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60F05",
      "60F15"
    ],
    "keywords": [
      "point process",
      "scale parameter",
      "boundary effects dominate",
      "two-component extreme value distribution",
      "random uniform points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}