On normal approximations to $U$-statistics

Let $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ be i.i.d. random observations. Let $\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge 2$ 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, $(\mathrm{var}\mathbb{T}){\ln }^2(\mathrm{var}\mathbb{T})$, under the condition $\mathbb{E}{\mathbb{T}}^2<\mathrm{\infty }$. An optimal bound without this $\log$ factor is obtained under a lower moment assumption ${\mathbb {E}|$\mathbb{T}$|^{\alpha}<\infty}$for${\alpha<2}$. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature.