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arXiv · 2610.16986

A quantitative CLT for the zeros of random trigonometric polynomials

Ao Huang·Zhonggen Su·Xiaolin Wang
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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.

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Ao Huang, Zhonggen Su, Xiaolin Wang. 2026-10-12. A quantitative CLT for the zeros of random trigonometric polynomials. https://arxiv.org/abs/2610.16986

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