{
  "abstract": "Let $X_1,X_2,\\ldots$ be i.i.d. random variables with zero mean and variance $σ^2\\in(0,\\infty)$. In this paper, we establish a sharp Cramér-type moderate deviation for the self-normalized maximum of partial sums under a finite $p$-th absolute moment, $2<p\\le3$. In particular, we show that $\\mathbb{P}(M_n\\ge xV_n)=2(1-Φ(x))\\{1+O(1)β_{n,p}(1+x)^p\\}$ for $0\\le x\\leβ_{n,p}^{-1/p}$, where $M_n=\\max_{1\\le k\\le n}\\sum_{j=1}^kX_j$, $V_n=(\\sum_{j=1}^nX_j^2)^{1/2}$, $β_{n,p}=\\mathbb{E}|X_1|^p/(n^{(p-2)/2}σ^p)$, and $Φ$ is the standard normal distribution function. This proves a conjecture of Shao2010. More precisely, we bound the relative error by the same truncated-moment quantity as for self-normalized sums. The proof compares the maximum with the absolute sum while preserving their common denominator, through a reflected-walk representation and Fourier estimates with distribution-adapted smoothing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21975",
  "author_total": 4,
  "authors": [
    {
      "name": "Song-Hao Liu"
    },
    {
      "name": "Qi-Man Shao"
    },
    {
      "name": "Hao Shi"
    },
    {
      "name": "Jing-Yu Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2bhmztaywz6zp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21975",
  "fulltext_url": null,
  "id": "2bhmztaywz6zp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21975",
  "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",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7222553820",
  "pdf_url": "https://arxiv.org/pdf/2610.21975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2bhmztaywz6zp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums",
  "title_text": "Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:30:00Z",
  "url": "https://research.draconitic.com/paper/2bhmztaywz6zp",
  "venue": "arXiv",
  "versions": []
}