发表机构
National University of Tainan; National Yang Ming Chiao Tung University（台南大学; 国立阳明交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究欧几里得球一维边缘分布的高斯逼近，建立尖锐渐近展开，证明在精确方差标准化下总变差距离以显式常数乘以N^{-1}衰减，从而确认先前O(N^{-1})界的最优性。
AI 中文摘要
我们研究通过归一化一维剖面函数得到的概率测度的高斯逼近，以一维欧几里得球的边缘分布为主要例子。我们首先在最大值集合附近建立定量集中估计。对于具有唯一非退化最大值点的剖面函数，我们给出充分条件，使得以最大值点为中心并根据剖面函数对数的局部二次逼近进行重新缩放后，所得密度在$L^1(\mathbb{R})$中收敛于标准高斯密度。然后我们专门研究$\mathbb{R}^n$中欧几里得球的一维边缘分布。记$N=n-1$，我们考虑边缘分布的两种标准化：一种由最大值点处的对数曲率决定，另一种由精确标准差决定。对于每种标准化，我们确定在$L^1(\mathbb{R})$中相对于标准高斯密度的$N^{-1}$阶首阶修正。这些展开式还确定了对称区间概率的相应首阶修正以及距标准高斯分布的总变差距离的首项。特别地，在精确方差标准化下，总变差距离渐近于一个显式正常数乘以$N^{-1}$。因此，先前已知的具有相同方差的高斯分布逼近的$O(N^{-1})$界在阶上是最优的。
英文摘要
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 $L^1(\mathbb{R})$ to the standard Gaussian density. We then specialize to one-dimensional marginals of Euclidean balls in $\mathbb{R}^n$. 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 $L^1(\mathbb{R})$. 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.
Commentskeywords:Profile measures, Euclidean balls, convex body sections, Gaussian approximation, marginal densities, first-order asymptotics, total variation
