Sharp Gaussian Asymptotics for Marginals of Euclidean Balls
Abstract
We study Gaussian approximation for probability measures obtained by normalizing one-dimensional profile functions, with one-dimensional marginals of Euclidean balls as the principal example. We first establish quantitative concentration estimates near the set of maximizers. For profiles with a unique nondegenerate maximizer, we give sufficient conditions under which centering at the maximizer and rescaling according to the local quadratic approximation of the logarithm of the profile yield densities that converge in L1(ℝ) to the standard Gaussian density. We then specialize to one-dimensional marginals of Euclidean balls in ℝn. Writing N=n−1, we consider two standardizations of the marginal distribution: one determined by the logarithmic curvature at the maximizer, and the other by the exact standard deviation. For each standardization, we identify the first-order correction, of order N−1, to the standard Gaussian density in L1(ℝ). These expansions also determine the corresponding first-order corrections to the probabilities of symmetric intervals and the leading terms of the total variation distances from the standard Gaussian distribution. In particular, under exact-variance standardization, the total variation distance is asymptotic to an explicit positive constant times N−1. Consequently, the previously known O(N−1) bound for approximation by the Gaussian distribution with the same variance is sharp in order.
Related
- Sharp L2-stability and gradient stability hierarchies of the L2-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality
- Small-Ball Marginals Do Not Control Restricted Eigenvalues by Euclidean Gaussian Width
- A sharp covering theorem and Solyanik estimates for Euclidean balls
- Revisiting the Bobkov-Chistyakov-Götze theorem on approximately Gaussian marginals
- Full-radius dimension-free maximal inequalities for discrete Euclidean balls
BibTeX
@misc{chen2026sharp,
title = {{Sharp Gaussian Asymptotics for Marginals of Euclidean Balls}},
author = {Bo-Si Chen and Yen-Chang Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14924},
eprint = {2609.14924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14924},
}
RIS
TY - UNPB
TI - Sharp Gaussian Asymptotics for Marginals of Euclidean Balls
AU - Bo-Si Chen
AU - Yen-Chang Huang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14924
UR - https://arxiv.org/abs/2609.14924
AB - We study Gaussian approximation for probability measures obtained by normalizing one-dimensional profile functions, with one-dimensional marginals of Euclidean balls as the principal example. We first establish quantitative concentration estimates near the set of maximizers. For profiles with a unique nondegenerate maximizer, we give sufficient conditions under which centering at the maximizer and rescaling according to the local quadratic approximation of the logarithm of the profile yield densities that converge in L^1(ℝ) to the standard Gaussian density. We then specialize to one-dimensional marginals of Euclidean balls in ℝ^n. Writing N=n−1, we consider two standardizations of the marginal distribution: one determined by the logarithmic curvature at the maximizer, and the other by the exact standard deviation. For each standardization, we identify the first-order correction, of order N^−1, to the standard Gaussian density in L^1(ℝ). These expansions also determine the corresponding first-order corrections to the probabilities of symmetric intervals and the leading terms of the total variation distances from the standard Gaussian distribution. In particular, under exact-variance standardization, the total variation distance is asymptotic to an explicit positive constant times N^−1. Consequently, the previously known O(N^−1) bound for approximation by the Gaussian distribution with the same variance is sharp in order.
ER -