A quantitative CLT for the zeros of random trigonometric polynomials

Abstract

Let $X_N(t)$, $t\in [0,\pi ]$, 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 $[0,N\pi ]$. 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.

arXiv comment31 pages