Three quantitative central limit theorems for the unbounded excursion component of a Gaussian field
Abstract
For a class of smooth stationary Gaussian fields, we study the fluctuations of the volume, surface area and Euler characteristic of the unbounded excursion component in growing boxes Λn=[−n,n]d. We establish nondegenerate central limit theorems for all three functionals throughout the supercritical regime in every dimension d≥ 2. We further quantify these central limit theorems by establishing Berry--Esseen bounds of order O(|Λn|−(1−1/s)) in Kolmogorov distance, where s∈(1,2] depend on the regularity and decay assumptions. Especially for the Bargmann--Fock field, we obtain a Berry--Esseen bound of O(|Λn|−1/2) for all three functionals. A matching lower bound shows that this rate is sharp for the Euler characteristic. For the volume, we improve the previously known rate of O(|Λn|−1/2(log|Λn|)2d). We mainly use Stein's method and moment bounds for differences arising from white-noise resampling to finish our proof.
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BibTeX
@misc{cheng2026three,
title = {{Three quantitative central limit theorems for the unbounded excursion component of a Gaussian field}},
author = {Yu Cheng and Song-Hao Liu and Qi-Man Shao and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37240},
eprint = {2610.37240},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37240},
}
RIS
TY - UNPB
TI - Three quantitative central limit theorems for the unbounded excursion component of a Gaussian field
AU - Yu Cheng
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Jing-Yu Xu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37240
UR - https://arxiv.org/abs/2610.37240
AB - For a class of smooth stationary Gaussian fields, we study the fluctuations of the volume, surface area and Euler characteristic of the unbounded excursion component in growing boxes Λ_n=[−n,n]^d. We establish nondegenerate central limit theorems for all three functionals throughout the supercritical regime in every dimension d≥ 2. We further quantify these central limit theorems by establishing Berry--Esseen bounds of order O(|Λ_n|^−(1−1/s)) in Kolmogorov distance, where s∈(1,2] depend on the regularity and decay assumptions. Especially for the Bargmann--Fock field, we obtain a Berry--Esseen bound of O(|Λ_n|^−1/2) for all three functionals. A matching lower bound shows that this rate is sharp for the Euler characteristic. For the volume, we improve the previously known rate of O(|Λ_n|^−1/2(log|Λ_n|)^2d). We mainly use Stein's method and moment bounds for differences arising from white-noise resampling to finish our proof.
ER -