## The Annals of Statistics

### The Berry-Esseen Theorem for $U$-Statistics

#### Abstract

Assuming only the existence of the third absolute moment we prove that $\sup_x |P(\sigma_n^{-1} U_n \leqq x) - \Phi (x)| \leqq C_{\nu_3\sigma_g}^{-3}n^{-\frac{1}{2}}$ where $U_n$ is a $U$-statistic. This concludes a series of investigations on the Berry-Esseen theorem for $U$-statistics by Grams and Serfling, Bickel, and Chan and Wierman.

#### Article information

Source
Ann. Statist., Volume 6, Number 2 (1978), 417-421.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344132

Digital Object Identifier
doi:10.1214/aos/1176344132

Mathematical Reviews number (MathSciNet)
MR464359

Zentralblatt MATH identifier
0393.60022

JSTOR
Berry-Esseen bound $U$-statistics
Callaert, Herman; Janssen, Paul. The Berry-Esseen Theorem for $U$-Statistics. Ann. Statist. 6 (1978), no. 2, 417--421. doi:10.1214/aos/1176344132. https://projecteuclid.org/euclid.aos/1176344132