{
  "abstract": "Recent work has shown that sub-Gaussian random variables are dominated in convex order by a sharp multiple of a Gaussian. We study the analogous question for the sub-Gamma class underlying Bernstein's inequality, with the Laplace law as the majorant. Here the moment generating function is controlled by a Bernstein-type bound over a finite range of frequencies, rather than by a purely quadratic bound. We derive a variational formula for the optimal multiple, show that it is strictly larger than the natural scale $σ\\vee α$, and prove that this value is sharp, being attained by an asymmetric two-point distribution. We then turn to the sub-exponential class, which has a quadratic bound as in the sub-Gaussian case, but only over a bounded range as in the sub-Gamma case. Interestingly, the sub-exponential class displays a different behavior: the optimal multiple is exactly $σ\\vee α$, but is attained only when $α\\leq σ$; when $α> σ$, the constant remains sharp, but equality cannot hold for any non-affine convex function. Both results rely on a sharp finite-range variant of the Kearns-Saul inequality, which is of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02398",
  "author_total": 2,
  "authors": [
    {
      "name": "El Mahdi Khribch"
    },
    {
      "name": "Badr-Eddine Chérief-Abdellatif"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9p7ce7n5t4j13.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02398",
  "fulltext_url": null,
  "id": "9p7ce7n5t4j13",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02398",
  "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": "W7207592338",
  "pdf_url": "https://arxiv.org/pdf/2609.02398",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9p7ce7n5t4j13.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02398",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02398"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convex Order Comparisons for Sub-Gamma Random Variables",
  "title_text": "Convex Order Comparisons for Sub-Gamma Random Variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/9p7ce7n5t4j13",
  "venue": "arXiv",
  "versions": []
}