Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 19 min ago
PreprintarXiv licencearXiv

A quantitative CLT for the zeros of random trigonometric polynomials

  • Ao Huang
  • Zhonggen Su
  • Xiaolin Wang
Open at sourcePDFDOICite

Abstract

Let XN(t), t∈[0,Ο€], be a stationary Gaussian trigonometric polynomial, and let 𝒡N denote the number of zeros of the rescaled process XN(t/N) on [0,NΟ€]. We establish the quantitative central limit theorem dW( (𝒡Nβˆ’π”Ό[𝒡N])/√(Var(𝒡N)),Z) ≀ C((log N)2)/√N, where Z is standard Gaussian and dW 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 Nβˆ’1/2 shows that the convergence rate is optimal up to logarithmic factors.

Related

  1. The Riemann zeta function at the zeros of real-rooted trigonometric polynomials
    Felipe Gonçalves, Bryce Kerr, Oleksiy Klurman +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  2. A note on the expectation of zeros of random harmonic polynomials: The Kac model
    Dawei Lu, Yuchen Wang
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  3. Central limit theorem for real zeros of random Weyl polynomials with general coefficients
    Ander Aguirre, Hoi H. Nguyen
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  4. Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials
    Feng Dai, Andriy Prymak
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsNumerical analysis & scientific computingarXiv
  5. Zeros of Ramanujan-Like Polynomials
    Benjamin Fichter, Vishal Shah, Wiseley Wong
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{huang2026quantitative,
  title = {{A quantitative CLT for the zeros of random trigonometric polynomials}},
  author = {Ao Huang and Zhonggen Su and Xiaolin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16986},
  eprint = {2610.16986},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16986},
}
RIS
TY  - UNPB
TI  - A quantitative CLT for the zeros of random trigonometric polynomials
AU  - Ao Huang
AU  - Zhonggen Su
AU  - Xiaolin Wang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16986
UR  - https://arxiv.org/abs/2610.16986
AB  - Let X_N(t), t∈[0,Ο€], be a stationary Gaussian trigonometric polynomial, and let 𝒡_N denote the number of zeros of the rescaled process X_N(t/N) on [0,NΟ€]. We establish the quantitative central limit theorem d_W( (𝒡_Nβˆ’π”Ό[𝒡_N])/√(Var(𝒡_N)),Z) ≀ C((log N)^2)/√N, where Z is standard Gaussian and d_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^βˆ’1/2 shows that the convergence rate is optimal up to logarithmic factors.
ER  -