{
  "abstract": "We study the completion time of a task subject to independent reset (catastrophe) at each step. The completion-time PGF depends on the base-process PGF through an affine relation, and we exploit this structure systematically. Our main result shows that, among age-based catastrophe mechanisms, geometric-tail catastrophe is exactly the class that yields uniform affine PGF structure; in continuous time, the characterization sharpens to Poisson resetting. We establish a sharp two-sided Kolmogorov bound of order $p+|α|$ for the exponential approximation $d_K(T/E[T], \\mathrm{Exp}(1))$, thereby closing a logarithmic gap. Applications to the coupon collector with reset coupons reveal a discontinuous Gumbel-to-Exponential transition under resetting, while a multi-phase model exhibits a Gaussian-to-exponential transition with exponential convergence rate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16566",
  "author_total": 2,
  "authors": [
    {
      "name": "Sichen Wang"
    },
    {
      "name": "Zhipeng Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3zx26ds7gbtas.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16566",
  "fulltext_url": null,
  "id": "3zx26ds7gbtas",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16566",
  "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",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7213417655",
  "pdf_url": "https://arxiv.org/pdf/2609.16566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3zx26ds7gbtas.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On Completion Times under Memoryless Catastrophe",
  "title_text": "On Completion Times under Memoryless Catastrophe",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/3zx26ds7gbtas",
  "venue": "arXiv",
  "versions": []
}