{
  "abstract": "We study extremal absolute central moments of Gaussian quadratic forms under a fixed Frobenius norm. For a nonzero real symmetric matrix $M$ and $G\\sim N(0,I_n)$, we construct an explicit centered difference of Gamma variables with the same mean, variance, and third centered moment as $G^{\\mathsf T}MG-\\operatorname{tr}M$. After normalization, replacing the quadratic form by this Gamma difference does not decrease $\\mathbb{E} f$ for every $C^2$ test function $f$ such that $f''$ is convex and $f$, $f'$, and $f''$ have polynomial growth. In particular, it gives an explicit upper bound for every absolute moment of order $p\\ge3$. We then prove that, for every $p\\ge4$, this bound is maximized by the centered square $g^2-1$ of a single standard Gaussian variable $g$. The resulting sharp inequality is \\[ \\bigl\\|G^{\\mathsf T}MG-\\operatorname{tr}M\\bigr\\|_p \\le \\|g^2-1\\|_p\\,\\|M\\|_{\\mathrm F}, \\] with equality if and only if $M$ has rank one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05914",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhekai Pang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/81ytxmnfttf2v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05914",
  "fulltext_url": null,
  "id": "81ytxmnfttf2v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05914",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7211998041",
  "pdf_url": "https://arxiv.org/pdf/2609.05914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/81ytxmnfttf2v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05914"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms",
  "title_text": "A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/81ytxmnfttf2v",
  "venue": "arXiv",
  "versions": []
}