发表机构
The Chinese University of Hong Kong, Shenzhen; Jinan University; South China University of Technology; National University of Singapore（香港中文大学（深圳）; 暨南大学; 华南理工大学; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对对称统计量建立最优正态逼近界，解决两个长期猜想，方法结合Prawitz平滑、条件方差子鞅、多项式逼近与有界测度变换。
AI 中文摘要
正态逼近中的收敛速率是概率论与统计学的基础。该理论已从标准化和推广到学生化统计量、样本均值的光滑函数以及$U$-统计量，并更一般地推广到对称统计量。在这一发展过程中，一个核心问题一直是确定在何种最弱假设下可以建立最优收敛速率。首先，在本文中，我们通过建立基于独立同分布观测的对称统计量的调整正态逼近的最优界来解决这一问题。更精确地，我们将Kolmogorov误差界定为经典三阶矩项与余项方差的一个非常简洁的和。这解决了Bentkus、Jing和Zhou（《概率年鉴》第37卷（2009年），第2174-2199页）提出的一个长期悬而未决的猜想。其次，我们在最小条件下，即第一投影的有限三阶绝对矩和正则核的有限$5/3$阶绝对矩下，建立了二阶刀切学生化$U$-统计量的最优Berry-Esseen速率。这解决了经典概率论中的一个长期悬而未决的猜想。我们的证明结合了带符号的Prawitz平滑、条件方差子鞅、多项式逼近以及有界测度变换。
英文摘要
Rates of convergence in normal approximation are fundamental to probability and statistics. The theory has evolved from normalized sums to Studentized statistics, smooth functions of sample means, and $U$-statistics, and more generally to symmetric statistics. A central question throughout this development has been to identify the weakest assumptions under which optimal rates of convergence can be established. First, in this paper we address this question by establishing optimal bounds for adjusted normal approximations of symmetric statistics based on independent and identically distributed observations. More precisely, we bound the Kolmogorov error by a very neat sum of the classical third-moment term and the variance of the remainder term. This resolves a long-standing conjecture posed by Bentkus, Jing and Zhou (\textit{Ann. Probab.} \textbf{37} (2009), 2174--2199). Secondly, we establish the optimal Berry-Esseen rate for jackknife Studentized \(U\)-statistics of order two under minimal conditions, i.e., a finite third absolute moment of the first projection and a finite \(5/3\)-absolute moment of the canonical kernel. This resolves a long-standing conjecture in classical probability theory. Our proofs combine signed Prawitz smoothing with a conditional-variance submartingale, polynomial approximation, and a bounded change of measure.
