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November 2009 On normal approximations to U-statistics
Vidmantas Bentkus, Bing-Yi Jing, Wang Zhou
Ann. Probab. 37(6): 2174-2199 (November 2009). DOI: 10.1214/09-AOP474


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


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Vidmantas Bentkus. Bing-Yi Jing. Wang Zhou. "On normal approximations to U-statistics." Ann. Probab. 37 (6) 2174 - 2199, November 2009.


Published: November 2009
First available in Project Euclid: 16 November 2009

zbMATH: 1186.62025
MathSciNet: MR2573555
Digital Object Identifier: 10.1214/09-AOP474

Primary: 62E20

Keywords: Berry–Esseen bound , central limit theorem , normal approximations , rate of convergence , Self-normalized , Studentized U-statistics , U-statistics

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 6 • November 2009
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