Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Huang, Ao, Su, Zhonggen, Wang, Xiaolin
Institutions: Zhejiang University, Chinese University of Hong Kong
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8898315279
OpenAlex ID: W7221828319