{
  "abstract": "We study the asymptotics as $n$ goes to infinity of $\\mathbb E\\left[\\mathcal{N}(S_n,[0,2π])\\right]$, the expected number of zeros in $[0, 2π]$ of a random periodic signal $S_n$ of the form \\[ S_n(t)=\\sum_{k=1}^{n}a_k f(kt), \\] where $f$ is a non-analytic $2π-$periodic function and the coefficients $(a_k)$ are i.i.d. random variables, centered with unit variance. We show in particular that if $a_1$ admits a finite third moment and if the function $f$ is piecewise polynomials and of class $\\mathcal C^{7}$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ \\[ \\lim_{n \\to +\\infty}\\frac{\\Esp\\left[\\mathcal{N}(S_n,[0,2π])\\right]}{n}= \\frac{2}{\\sqrt{3}}\\sqrt{\\frac{\\|f'\\|_{L^2([0,2π])}}{\\|f\\|_{L^2([0,2π])}}}. \\] This result thus extends in expectation and at the scale of the whole period $[0,2π]$ the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size $1/n$. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem à la Salem--Zygmund for the function $S_n$ when evaluated at a uniform random point in $[0, 2π]$, and as well as suitable uniform integrability and anti-concentration estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01007",
  "author_total": 3,
  "authors": [
    {
      "name": "Jürgen Angst"
    },
    {
      "name": "Thibault Pautrel"
    },
    {
      "name": "Guillaume Poly"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dv9xabf545ryv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01007",
  "fulltext_url": null,
  "id": "dv9xabf545ryv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01007",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7206173686",
  "pdf_url": "https://arxiv.org/pdf/2609.01007",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dv9xabf545ryv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01007",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01007"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Global universality of the expected number of zeros of non-analytic random signals",
  "title_text": "Global universality of the expected number of zeros of non-analytic random signals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/dv9xabf545ryv",
  "venue": "arXiv",
  "versions": []
}