Optimal bounds in non-Gaussian limit theorems for \(U\)-statistics
type: Item
instance of: scholarly article
title: Optimal bounds in non-Gaussian limit theorems for \(U\)-statistics
zbMATH Open document ID: 1008.62017
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: It is known that \(U\)-statistics given by symmetric kernels of degree two, asymptotically behave like weighted quadratic forms. The limit variable turns out to be an infinite weighted sum of squares of i.i.d. standard normal random variables which produce the non-Gaussian limit. The present paper establishes an Edgeworth expansion for that type of \(U\)-statistics. Explicit bounds of order \(O(N^{-1})\) for the uniform error of the distribution functions are derived. The results are related to a special representation of \(U\)-statistics, which is based on a related series of independent Gaussian random variables. The approach is based on a spectral decomposition of the underlying kernel.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1359638
zbMATH Keywords: U-statistics
zbMATH Keywords: degenerate U-statistics
zbMATH Keywords: von Mises statistics
zbMATH Keywords: symmetric statistics
zbMATH Keywords: central limit theorem
zbMATH Keywords: convergence rates
zbMATH Keywords: Berry-Esseen bounds
zbMATH Keywords: Edgeworth expansions
zbMATH Keywords: second order efficiency
author: Vidmantas Bentkus
author: Friedrich Götze
reviewed by: Arnold Janssen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677269
OpenAlex ID: W2052239055
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DOI: 10.1214/AOP/1022677269
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description: scientific article; zbMATH DE number 1359638