On exact rates of convergence in functional limit theorems for \(U\)- statistic type processes
type: Item
instance of: scholarly article
title: On exact rates of convergence in functional limit theorems for \(U\)- statistic type processes
zbMATH Open document ID: 0813.60020
DOI: 10.1007/BF02214367
author: Dietmar Ferger
published in: Journal of Theoretical Probability
publication date: 1994-11-14T00:00:00Z
review text: Let \(X,X_ 1, X_ 2, \dots\) be an i.i.d. sequence of random variables taking values in a measurable space. Let \(K(x,y)\) be a symmetric (either antisymmetric) measurable function. Consider the partial sums \(\sum^ n_{i=k+1} \sum^ k_{j=1} K(X_ i,X_ j)\), for integers \(0 \leq k \leq n\), related to the \(U\)-statistics kernel function \(K\). Similarly to the invariance principle, continuous random polygonal lines may be defined. Under certain conditions the random \(U\)-statistic type process with the polygonal lines as sample functions converges in distribution as \(n \to \infty\) to a Gauassian process related to the Brownian motion. The author obtains Berry-Esseen bounds for the convergence rate of these processes in the Prokhorov distance. The optimality of bounds is shown. Applications for the behavior of changepoint estimators are given. Actually the author considers the more general situation of triangular rowwise i.i.d. arrays \(X_{1n}, \dots, X_{nn}\). The main idea of the proof is a reduction to certain partial sum processes.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 687642
zbMATH Keywords: Berry-Esseen bound
zbMATH Keywords: \(U\)-statistics and processes
zbMATH Keywords: invariance principle
zbMATH Keywords: Hungarian construction
zbMATH Keywords: changepoint estimators
zbMATH Keywords: Prokhorov distance
reviewed by: Friedrich Götze
MaRDI profile type: Publication
cites work: On an Estimate of A. A. Borovkov
cites work: On the Rate of Convergence for the Invariance Principle
cites work: Q4114551
cites work: Invariance principles for changepoint problems
cites work: Convergence of changepoint estimators
cites work: An exact rate of convergence in the functional central limit theorem for special martingale difference arrays
cites work: Probability Inequalities for Sums of Bounded Random Variables
description: scientific article; zbMATH DE number 687642