On a normal approximation of $U$-statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 3, pp. 469-488
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We consider $U$-statistics of order 2 constructed upon independent identically distributed random variables $X_1,\ldots,X_n$ with values in a measurable space $(\mathfrak{X,B})$. For $U$-statistics with a nondegenerate kernel and canonical functions $g\colon \mathfrak{X}\mapsto\mathbf{R}$ and $h\colon \mathfrak{X}^2\mapsto\mathbf{R}$, we investigate a problem on the estimation of the rate of convergence in the central limit theorem. The result obtained implies that the estimate of order $n^{-1/2}$ depends only on the third moment $\mathbf{E}|g(X_1)|^3$ and the weak moment $\sup_{x > 0}(x^{5/3} \mathbf{P}\{|h(X_1,\,X_2)| > x\})$ of order ${\frac{5}{3}}$.
Keywords:
$U$-statistic, normal approximation, Berry–Esséen inequality, central limit theorem.
@article{TVP_2000_45_3_a2,
author = {Yu. V. Borovskikh},
title = {On a normal approximation of $U$-statistics},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {469--488},
publisher = {mathdoc},
volume = {45},
number = {3},
year = {2000},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2000_45_3_a2/}
}
Yu. V. Borovskikh. On a normal approximation of $U$-statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 3, pp. 469-488. http://geodesic.mathdoc.fr/item/TVP_2000_45_3_a2/