Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials
Abstract
We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials Pn, where n denotes the degree and is assumed to be sufficiently large. For each fixed B above an absolute threshold, let In=[−√n+B√(log n), √n−B√(log n)]. Uniformly over deterministic compact intervals I⊆ In whose length ℓ is sufficiently large and depends on n, the distribution of the standardized number of real roots in I has Kolmogorov distance at most Clogℓ/√ℓ from the standard Gaussian distribution. Consequently, every such interval sequence with ℓ→∞ satisfies a central limit theorem. In particular, taking I=In gives the bound Clog n/n1/4. The same bound holds for the distribution of the standardized number of real roots on ℝ. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.
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BibTeX
@misc{wang2026berryesseen,
title = {{Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials}},
author = {Yuchen Wang and Dawei Lu and Song-Hao Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12734},
eprint = {2609.12734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12734},
}
RIS
TY - UNPB
TI - Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials
AU - Yuchen Wang
AU - Dawei Lu
AU - Song-Hao Liu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12734
UR - https://arxiv.org/abs/2609.12734
AB - We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials P_n, where n denotes the degree and is assumed to be sufficiently large. For each fixed B above an absolute threshold, let I_n=[−√n+B√(log n), √n−B√(log n)]. Uniformly over deterministic compact intervals I⊆ I_n whose length ℓ is sufficiently large and depends on n, the distribution of the standardized number of real roots in I has Kolmogorov distance at most Clogℓ/√ℓ from the standard Gaussian distribution. Consequently, every such interval sequence with ℓ→∞ satisfies a central limit theorem. In particular, taking I=I_n gives the bound Clog n/n^1/4. The same bound holds for the distribution of the standardized number of real roots on ℝ. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.
ER -