{
  "abstract": "Let $X_1,X_2, \\ldots $ be independent uniform random points in a bounded region $A \\subset {\\bf R}^d$ having a smooth boundary, $d \\geq 1$. Let $B \\subset A$ be compact. The _coverage threshold_ of $B$, $R_n$, is the smallest $r$ such that $B$ is covered by the balls of radius $r$ centred on $X_1,\\ldots,X_n$. The _maximal spacing_ $\\tilde{R}_n$ is the volume of the largest ball contained in $A \\setminus \\{X_1,\\ldots,X_n\\}$. We investigate the asymptotic frequency of _record times_ in the sequence $(R_n)$, that is times $n$ for which $R_n < R_{n-1}$. Let $N_m$ denote the number of records in the sequence $(R_n)$ up to time $m$, and let $ν_m$ be the time at which the $m$th record value of the sequence $(R_n)$ occurs. For $B \\subset A^o$, we show that almost surely, $N_n \\sim \\frac12 (\\log n)^2$ and $ν_n^{1/\\sqrt{n}}\\to \\exp \\big(\\sqrt{2}\\: \\big)$ as $n \\to \\infty$, and likewise for $\\tilde{N}_n$ and $\\tildeν_n$, defined analogously in terms of $(\\tilde{R}_n)$. But if $B=A$ and $d \\geq 3$, then $N_n \\sim \\frac12 (1- \\frac{1}{d}) (\\log n)^2$ and $ν_n^{1/\\sqrt{n}}\\to \\exp \\big( \\sqrt{2d/(d-1)} \\: \\big)$. We also discuss the generalization (for fixed $k \\in {\\bf N}$) to $k$-coverage thresholds, maximal $k$-spacings and non-uniformly distributed points $X_i$ in $A$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19104",
  "author_total": 1,
  "authors": [
    {
      "name": "Mathew D. Penrose"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cz43grh0ja4j2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19104",
  "fulltext_url": null,
  "id": "cz43grh0ja4j2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19104",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7203870420",
  "pdf_url": "https://arxiv.org/pdf/2608.19104",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cz43grh0ja4j2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2608.19104",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19104"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Record times for coverage thresholds and maximal spacings",
  "title_text": "Record times for coverage thresholds and maximal spacings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/cz43grh0ja4j2",
  "venue": "arXiv",
  "versions": []
}