Optimal normal approximation for symmetric statistics
Abstract
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.
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BibTeX
@misc{jing2026optimal,
title = {{Optimal normal approximation for symmetric statistics}},
author = {Bing-Yi Jing and Yiming Liu and Shaochen Wang and Wang Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02724},
eprint = {2610.02724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02724},
}
RIS
TY - UNPB
TI - Optimal normal approximation for symmetric statistics
AU - Bing-Yi Jing
AU - Yiming Liu
AU - Shaochen Wang
AU - Wang Zhou
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02724
UR - https://arxiv.org/abs/2610.02724
AB - 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.
ER -