On normal approximations to U-statistics

Abstract

Let X₁, …, X_n be i.i.d. random observations. Let ${\mathbb{S}=\mathbb{L}+\mathbb{T}}$ be a U-statistic of order k≥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, $(\operatorname{var}\mathbb{T})\ln^{2}(\operatorname{var}\mathbb{T})$, under the condition ${\mathbb{E}\mathbb{T}^{2}<\infty}$. 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.