{
  "abstract": "Let $X_N(t)$, $t\\in[0,π]$, be a stationary Gaussian trigonometric polynomial, and let $\\mathcal Z_N$ denote the number of zeros of the rescaled process $X_N(t/N)$ on $[0,Nπ]$. We establish the quantitative central limit theorem \\[ d_{\\mathrm W}\\left( \\frac{\\mathcal Z_N-\\mathbb E[\\mathcal Z_N]} {\\sqrt{\\operatorname{Var}(\\mathcal Z_N)}},Z\\right) \\le C\\frac{(\\log N)^2}{\\sqrt N}, \\] where $Z$ is standard Gaussian and $d_{\\mathrm W}$ denotes the Wasserstein-1 distance. The proof combines Ornstein--Uhlenbeck regularization of the singular Kac representation with direct estimates of the covariance between the Malliavin--Stein factor and Stein test functions. A lattice lower bound of order $N^{-1/2}$ shows that the convergence rate is optimal up to logarithmic factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16986",
  "author_total": 3,
  "authors": [
    {
      "name": "Ao Huang"
    },
    {
      "name": "Zhonggen Su"
    },
    {
      "name": "Xiaolin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/emywhmxdj3xmf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16986",
  "fulltext_url": null,
  "id": "emywhmxdj3xmf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16986",
  "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": "W7221828319",
  "pdf_url": "https://arxiv.org/pdf/2610.16986",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/emywhmxdj3xmf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:16Z",
      "native_id": "2610.16986",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16986"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A quantitative CLT for the zeros of random trigonometric polynomials",
  "title_text": "A quantitative CLT for the zeros of random trigonometric polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:16Z",
  "url": "https://research.draconitic.com/paper/emywhmxdj3xmf",
  "venue": "arXiv",
  "versions": []
}