Central limit theorem for real zeros of random Weyl polynomials with general coefficients

Abstract

For a random polynomial, the number of real zeros $N_{\mathbb R}$ is a highly nonlinear function of its coefficients, and its statistical properties have been studied extensively. A natural question is whether $N_{\mathbb R}$ satisfies a central limit theorem. For various ensembles with iid standard Gaussian coefficients, such central limit theorems have been established; see, for instance, arXiv:1911.12182, arXiv:1801.06331, arXiv:1401.5745; Azais and Leon, Electron. J. Probab. 18 (2013), no. 68; arXiv:1504.05355, arXiv:2111.09015, arXiv:1707.09276, arXiv:1005.4113. These results rely on a rich range of tools, including Kac-Rice formulas, moment methods, and Wiener chaos decompositions. In the non-Gaussian setting, however, many of these tools are unavailable. To the best of our knowledge, prior central limit theorems beyond the Gaussian setting were limited to Kac-type polynomials, including hyperbolic polynomials; see the works of Maslova (1974), O. Nguyen and Vu (arXiv:1904.04347), and, more recently, Do, N. Nguyen, and O'Rourke (arXiv:2605.26402). In this paper, we prove a central limit theorem for the total number of real zeros of Weyl polynomials whose coefficients are iid copies of a symmetric, mean-zero, variance-one subgaussian random variable $\xi$. This substantially extends one of the main results of Do and Vu (arXiv:1707.09276) to a broad class of non-Gaussian distributions, including the Rademacher distribution. Without the symmetry assumption, we prove central limit theorems for the number of real zeros for positive bulk intervals, as well as for $[0,\infty)$. Our proof combines the uniform one-point anti-concentration estimates from our recent work (arXiv:2511.07735) with the localization of Weyl polynomials around the coefficient index $i\approx x^2$. While our proofs use comparison to compute the variances, the CLT deduction is rather direct.