Bounds for Non-Gaussian Approximations of U-Statistics期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Bounds for Non-Gaussian Approximations of U-Statistics

Authors:

Affiliation:

(1) Department of Applied Mathematics, Transport University, Moskovsky Avenue 9, 190031 St. Petersburg, Russia;(2) School of Mathematics and Statistics, F07, University of Sydney, N.S.W, 2006, Australia

Abstract:

Let X,

,X₁,...,X_n be i.i.d. random variables taking values in a measurable space ( $${mathfrak{X}},{mathfrak{B}}$$

). Consider U-statistics of degree two

$U_n = n^{ - 1} mathop sum limits_{1 leqslant i < j leqslant n} Phi (X_i ,X_j )$

with symmetric, degenerate kernel $$Phi :{mathfrak{X}} mapsto {mathbb{R}}$$

. Let

$U_infty = frac{1}{2}sumlimits_{j = 1}^infty {q_j (tau } _j^2 - 1)$

where {q_j} are eigenvalues of the Hilbert–Schmidt operator associated with the kernel and {_j} are i.i.d. standard normal random variables. If ${mathbb{E}}Phi ^2 (X,bar X) < infty$ then

$Delta _n : = mathop {sup }limits_x |mathbb{P}{ U_n leqslant x} - mathbb{P}{ U_infty leqslant x} | to {text{ }}0{text{ as }}n to infty$

Upper bounds for _n are established under the moment condition $mathbb{E}Phi ^2 (X,bar X) = 1$ , provided that at least thirteen eigenvalues of the operator do not vanish. In Theorem 1.1 the bound is expressed via terms containing tail and truncated moments. The proof is based on the method developed by Bentkus and Götze.⁽¹⁾

Keywords:

U-statistics U-statistical sums symmetric degenerate kernel Gaussian random variables tail moments

本文献已被 SpringerLink 等数据库收录！

设为首页 | 免责声明 | 关于勤云 | 加入收藏