{
  "abstract": "We solve the problem posed by Erdős and Rényi in $1961$ about the limiting distribution of the number of draws $T_{n,m}$ that a coupon collector needs to complete $m$ collections of $n$ distinct types of coupons, when $m$ increases with $n$. For fixed $m$, Erdős and Rényi proved that $\\mathbb P(T_{n,m}\\leq n\\log n+(m-1)n\\log\\log n-n\\log (m-1)!+θn)$ converges to $ \\mathcal G(θ)$, for every $θ\\in \\mathbb R$, as $n \\to \\infty$, where $\\mathcal G$ is the standard Gumbel distribution function, and they asked what happens when $m=m(n)$ increases with $n$. We give a complete answer to this question. More precisely, for every increasing sequence $m=m(n)$, we determine explicit sequences $t_{n,m}$ and $w_{n,m}$ satisfying $w_{n,m}/t_{n,m}\\to0$ such that $\\mathbb P(T_{n,m}\\leq t_{n,m}+θw_{n,m})\\to\\mathcal G(θ)$ for every $θ\\in\\mathbb R$. We also determine the corresponding asymptotic behavior of $\\mathbb E[T_{n,m}]$. To prove these results, we present a methodology, inspired by Poissonization, that can be applied to determine the asymptotic behaviors of standardized hitting times of discrete-time stochastic processes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33199",
  "author_total": 2,
  "authors": [
    {
      "name": "Maria Eliza Castro"
    },
    {
      "name": "Rodrigo Marinho"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ewpza6nv1sd5y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33199",
  "fulltext_url": null,
  "id": "ewpza6nv1sd5y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33199",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7223918167",
  "pdf_url": "https://arxiv.org/pdf/2610.33199",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ewpza6nv1sd5y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33199",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33199"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Cutoff profile for the coupon collector with many collections",
  "title_text": "Cutoff profile for the coupon collector with many collections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/ewpza6nv1sd5y",
  "venue": "arXiv",
  "versions": []
}