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This article is cited in 8 scientific papers (total in 8 papers)
On a normal approximation of $U$-statistics
Yu. V. Borovskikh Petersburg State Transport University
Abstract:
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.
Received: 17.12.1997 Revised: 24.11.1998
Citation:
Yu. V. Borovskikh, “On a normal approximation of $U$-statistics”, Teor. Veroyatnost. i Primenen., 45:3 (2000), 469–488; Theory Probab. Appl., 45:3 (2001), 406–423
Linking options:
https://www.mathnet.ru/eng/tvp480https://doi.org/10.4213/tvp480 https://www.mathnet.ru/eng/tvp/v45/i3/p469
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