Let ${X_1,...,X_n}$ be i.i.d. random observations. Let $\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge2$ where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb{T}$ is a stochastically smaller statistic. We show that the rate of convergence to normality for $\mathbb{S}$ can be simply expressed as the rate of convergence to normality for the linear part $\mathbb{L}$ plus a correction term, $(\operatorname {var}\mathbb{T})\ln^2(\operatorname {var}\mathbb{T})$, under the condition ${\mathbb{E}\mathbb{T}^2

Published in at http://dx.doi.org/10.1214/09-AOP474 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)