{
  "abstract": "Let $X,X_1,X_2,\\ldots$ be i.i.d.\\ random vectors in $\\mathbb R^d$, where $d\\ge1$ is fixed. Assume that $\\mathbb{E} X=0$, $\\mathbb{E} XX^\\top=I_d$ and $\\mathbb{E}\\|X\\|^3<\\infty$. We prove a Cramér type moderate deviation theorem for a multivariate self-normalized radius statistic, that is, for every $a_n=o(n^{1/6})$, \\[ \\sup_{0\\le x\\le a_n} \\left|\\frac{\\mathbb{P}(U_{n,+}\\ge x)}{\\mathbb{P}(χ_d\\ge x)}-1\\right|\\to0, \\] where $χ_d$ is the Euclidean norm of a standard normal vector in $\\mathbb{R}^d$ and \\[ U_{n,+}^2=S_n^\\top A_n^+S_n, \\qquad S_n=\\sum_{j=1}^nX_j, \\qquad A_n=\\sum_{j=1}^nX_jX_j^\\top. \\] Here $A_n^+$ denotes the Moore--Penrose inverse. We further prove that the same result holds for the corresponding Hotelling $T^2$ statistic. This work closes the moment gap between the previously known finite $(3+δ)$ moment condition for Hotelling moderate deviations and the finite third moment boundary. Our proof adopts a geometric approach. Truncation at the finite-third-moment scale yields a truncated covariance matrix whose concentration error is negligible relative to the target \\(χ_d\\) tail. Uniform projected self-normalized moderate deviations are then averaged over directions and, after a whitening map with a ridge term, become a spherical cap transform of the radial distribution. A local Abel inversion recovers the radial tail, and deterministic comparison removes the ridge and the truncation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05822",
  "author_total": 4,
  "authors": [
    {
      "name": "Bing-Yi Jing"
    },
    {
      "name": "Yiming Liu"
    },
    {
      "name": "Shaochen Wang"
    },
    {
      "name": "Wang Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0kdq4q295qpx7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05822",
  "fulltext_url": null,
  "id": "0kdq4q295qpx7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05822",
  "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"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05822",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0kdq4q295qpx7.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05822",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05822"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A geometric proof of multivariate self-normalized moderate deviations via ridge regularization",
  "title_text": "A geometric proof of multivariate self-normalized moderate deviations via ridge regularization",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:32Z",
  "url": "https://research.draconitic.com/paper/0kdq4q295qpx7",
  "venue": "arXiv",
  "versions": []
}