{
  "abstract": "Let $A=(A_n)_{n\\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\\le i,j\\le n}$ is an $n\\times n$ random matrix with independent real entries satisfying $\\mathbb E a_{ij}=0$ and $\\mathbb Ea_{ij}^2=1$, and put $\\mathcal L_n=\\log|\\det A_n|$ and \\[ W_n^{\\mathrm d}(A_n):=\\frac{\\mathcal L_n - \\frac12\\log(n-1)!}{\\sqrt{\\frac12\\log n}},\\quad W_n^{\\mathrm e}(A_n):= \\frac{\\mathcal L_n-\\mathbb E \\mathcal L_n}{\\sqrt{\\frac12\\log n}}. \\] We prove that $W_n^{\\mathrm d}(A_n) \\Rightarrow \\mathcal N(0,1)$, whenever the family $\\left\\{\\frac{|a_{ij}|^{4}}{\\sqrt{\\log(e+|a_{ij}|)}} \\right\\}_{n\\geq 2;1\\leq i,j\\leq n}$ is uniformly integrable. If, in addition, the entries have uniformly bounded densities, then $W_n^{\\mathrm e}(A_n) \\Rightarrow \\mathcal N(0,1)$ whenever the family $\\left\\{\\frac{|a_{ij}|^{4}}{\\log(e+|a_{ij}|)}\\right\\}_{n\\geq 2;1\\leq i,j\\leq n} $ is uniformly integrable. These two conditions are optimal at the level of universal moment assumptions. We further establish the corresponding Berry--Esseen bounds, and show that for $0<δ\\le\\tfrac12$, if $\\sup\\limits_{n}\\max\\limits_{1\\le i,j\\le n}\\mathbb E \\frac{|a_{ij}|^4} {\\{\\log(e+|a_{ij}|)\\}^{1/2-δ}}<\\infty$, then \\begin{align*} d_{\\mathrm K}(W_n^{\\mathrm d}(A_n),\\mathcal N(0,1))\\le C(\\log n)^{-δ}. \\end{align*} For $0<γ\\le1$, if $\\sup\\limits_{n}\\max\\limits_{1\\le i,j\\le n}\\mathbb E \\frac{|a_{ij}|^4} {\\{\\log(e+|a_{ij}|)\\}^{1-γ}}<\\infty$ and the entries have uniformly bounded densities, then \\begin{align*} d_{\\mathrm K}(W_n^{\\mathrm e}(A_n),\\mathcal N(0,1))\\le C(\\log n)^{-γ}. \\end{align*} When $δ= 1/2$ and $γ= 1$, the bounds $(\\log n)^{-1/2}$ and $(\\log n)^{-1}$ are optimal, respectively. Our results improve the earlier Central Limit Theorem by \\cite{BaoPanZhou2015} and the Berry--Esseen bound by \\cite{NguyenVu2014}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11688",
  "author_total": 3,
  "authors": [
    {
      "name": "Song-Hao Liu"
    },
    {
      "name": "Qi-Man Shao"
    },
    {
      "name": "Jing-Yu Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmdjx0y462vzz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11688",
  "fulltext_url": null,
  "id": "dmdjx0y462vzz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11688",
  "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|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212286198",
  "pdf_url": "https://arxiv.org/pdf/2609.11688",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmdjx0y462vzz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11688",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11688"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants",
  "title_text": "The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/dmdjx0y462vzz",
  "venue": "arXiv",
  "versions": []
}