On normal approximations to $U$-statistics

Details On normal approximations to $U$-statistics On normal approximations to $U$-statistics

2009-03-18

arxiv.org/abs/0903.3081v2

Annals of Probability 2009, Vol. 37, No. 6, 2174-2199

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP474 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP474

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.

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