{
  "abstract": "Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \\[ \\mathbb{P}\\left[\\sum_{i=1}^n X_i\\ge t\\right] \\leq 1-\\left(1-\\frac{1}{t}\\right)^n \\text{ for all } t\\ge 2n+1 \\,. \\] To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\\mathbb{P}\\left[X_i=0\\right]= 1-\\frac{1}{t}$ and $\\mathbb{P}\\left[X_i=t\\right]=\\frac{1}{t}$, which remains the maximizer in the relaxed problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06317",
  "author_total": 2,
  "authors": [
    {
      "name": "Philipp Strack"
    },
    {
      "name": "Jannik M. Westermann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5bf5k7nvk8w8z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06317",
  "fulltext_url": null,
  "id": "5bf5k7nvk8w8z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06317",
  "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": "W7201867507",
  "pdf_url": "https://arxiv.org/pdf/2608.06317",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5bf5k7nvk8w8z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.06317",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06317"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Tail Bounds Beyond Twice the Mean",
  "title_text": "Sharp Tail Bounds Beyond Twice the Mean",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/5bf5k7nvk8w8z",
  "venue": "arXiv",
  "versions": []
}