The Annals of Probability

On normal approximations to U-statistics

Vidmantas Bentkus, Bing-Yi Jing, and Wang Zhou

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Let X1, …, Xn 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 ${\mathbb{E}\mathbb{T}^{2}\textless \infty}$. An optimal bound without this log factor is obtained under a lower moment assumption ${\mathbb{E}|\mathbb{T}|^{\alpha}\textless \infty}$ for ${\alpha \textless 2}$. 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.

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Ann. Probab., Volume 37, Number 6 (2009), 2174-2199.

First available in Project Euclid: 16 November 2009

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Primary: 62E20: Asymptotic distribution theory

U-statistics Berry–Esseen bound rate of convergence central limit theorem normal approximations self-normalized Studentized U-statistics


Bentkus, Vidmantas; Jing, Bing-Yi; Zhou, Wang. On normal approximations to U -statistics. Ann. Probab. 37 (2009), no. 6, 2174--2199. doi:10.1214/09-AOP474.

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