On normal approximations to $U$-statistics

TL;DR

This paper investigates the rate at which $U$-statistics converge to a normal distribution, providing explicit bounds and conditions under which the convergence rate can be optimized, extending existing theoretical results.

Contribution

It derives simple and optimal bounds for the normal approximation of $U$-statistics, including correction terms and conditions for improved convergence rates.

Findings

Rate of convergence expressed as sum of linear part and correction term

Optimal bounds without logarithmic factor under lower moment assumptions

Extension and refinement of existing results on $U$-statistics normal approximation

Abstract

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 ${\mathbb{E}\mathbb{T}^2<\infty}$. 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.