Optimal rates of convergence of second-degree polynomials in several metrics
type: Item
instance of: scholarly article
title: Optimal rates of convergence of second-degree polynomials in several metrics
zbMATH Open document ID: 1119.60017
author: O. L. Yanushkyavichene
published in: Journal of Mathematical Sciences (New York)
publication date: 2007-04-19T00:00:00Z
review text: Let \(X_1, X_2,\dots\) be a sequence of zero mean, independent and identically distributed random variables with finite fourth moments. Assume the variance of \(X_1\) is 1. Let \(Y_k = X_k^2-1\), \(T_1 = n^{-1}(Y_1+\dots+Y_n)\) and \(T_2=n^{-1/2}(X_1+ \dots + X_n)\). The paper considers the rate at which \(T=T_2^2 - T_1 -1\) converges to a mean adjusted \(\chi^2_1\) distribution as measured by different metrics. In particular the optimal rate of convergence is obtained for the Kolmogorov distance and for the Lévy distance in the sense that the rates are achieved in the case where the \(X_i\) have a standard normal distribution.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 5144824
zbMATH Keywords: degenerate U-statistics
zbMATH Keywords: rates of convergence
zbMATH Keywords: Lévy distance
zbMATH Keywords: Kolmogorov distance
zbMATH Keywords: polynomial
reviewed by: Neville Weber
MaRDI profile type: Publication
full work available at URL: s10958-006-0314-4
OpenAlex ID: W1990828901
cites work: On the dependence of the Berry-Esseen bound on dimension
cites work: On the rate of convergence of second-degree random polynomials
cites work: Modern Theory of Summation of Random Variables
DOI: 10.1007/S10958-006-0314-4
description: scientific article; zbMATH DE number 5144824