{
  "abstract": "Let $0\\le μ_1\\le μ_2 \\le \\cdots \\le μ_n$ and $δ> 0$. Samuels' conjecture states that if $X_1,\\dots,X_n$ are independent non-negative random variables with $\\mathbb{E}[X_i] = μ_i$, then $$ \\mathbb{P}\\left( \\sum_{i=1}^n X_i < δ+ \\sum_{i=1}^n μ_i \\right) \\ge \\min_{1\\le i\\le n} \\prod_{j=i}^n \\left(1-\\frac{μ_j}{δ+ \\sum_{k=i}^n μ_k}\\right).$$ This conjecture was recently proved by Ling. In this note, we prove a refinement of Samuels' inequality via an operation called the \\emph{cap move}. Our refined inequality is sharp, with equality attained for a strictly larger class of distributions than in the original inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22816",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanjun Han"
    },
    {
      "name": "Zipei Nie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/etn8ag0breyw7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22816",
  "fulltext_url": null,
  "id": "etn8ag0breyw7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22816",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7204166555",
  "pdf_url": "https://arxiv.org/pdf/2608.22816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/etn8ag0breyw7.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2608.22816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Sharp Refinement of Samuels' Inequality via Cap Moves",
  "title_text": "A Sharp Refinement of Samuels' Inequality via Cap Moves",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/etn8ag0breyw7",
  "venue": "arXiv",
  "versions": []
}