{
  "id": "2105.00320",
  "version": "v1",
  "published": "2021-05-01T18:27:50.000Z",
  "updated": "2021-05-01T18:27:50.000Z",
  "title": "Gaussian approximation in random minimal directed spanning trees",
  "authors": [
    "Chinmoy Bhattacharjee"
  ],
  "comment": "24 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the total length of rooted edges in a random minimal directed spanning tree - first introduced in \\cite{BR04} - in the unit cube $[0,1]^d$ for $d \\ge 3$. In the case of $d=2$, a Dickman limit was proved in \\cite{PW04}. This raises the question, what happens in dimensions three and higher and is it possible to prove a quantitative limit theorem? In this paper, we prove a Gaussian central limit theorem for $d \\ge 3$. Moreover, by utilizing recent results from \\cite{BM21} originating in Stein's method, we provide presumably optimal non-asymptotic bounds on the Wasserstein and the Kolmogorov distances between the distributions of total length of rooted edges, suitably normalized, and a standard Gaussian random variable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-01T18:27:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60D05"
    ],
    "keywords": [
      "random minimal directed spanning tree",
      "gaussian approximation",
      "gaussian central limit theorem",
      "total length",
      "presumably optimal non-asymptotic bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}