Weak invariance principles for weighted \(U\)-statistics
type: Item
instance of: scholarly article
title: Weak invariance principles for weighted \(U\)-statistics
zbMATH Open document ID: 0790.60036
DOI: 10.1007/BF02213365
published in: Journal of Theoretical Probability
publication date: 1994-06-20T00:00:00Z
review text: This paper deals with the weak limit behaviour of weighted, degenerate \(U\)-statistics \(U_ n=\sum_ n'a_ Ih(X_ I)\), where \(\sum_ n'\) denotes the sum over all multi-indices \(I=(i_ 1,\dots,i_ m)\in\{1,\dots,n\}^ m\), \((a_ I)\) is a symmetric array of real numbers and \(X_ I=(X_{i_ 1},\dots,X_{i_ m})\), where \(X_ 1,X_ 2,\dots\) is a sequence of independent and identically distributed (i.i.d.) random variables. The distribution of \(U_ n\) is shown to be close, in the Lévy-Prokhorov metric, to the distribution of a certain function \(V_ n\), in i.i.d. normal random variables. For the case \(m\geq 2\) a Rotar'-type invariance principle for degenerate \(U\)-statistics is established.
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 501135
zbMATH Keywords: weighted \(U\)-statistics
zbMATH Keywords: incomplete \(U\)-statistics
zbMATH Keywords: degenerate \(U\)- statistics
zbMATH Keywords: invariance principle
author: T. Mikosch
reviewed by: Neville Weber
MaRDI profile type: Publication
cites work: On the bootstrap of \(U\) and \(V\) statistics
cites work: Some properties of incomplete U-statistics
cites work: Martingale Central Limit Theorems
cites work: Reduced U-statistics and the Hodges-Lehmann estimator
cites work: Resampling \(U\)-statistics using \(p\)-stable laws
cites work: Invariance principles for von Mises and U-statistics
cites work: Q3756303
cites work: Q5336643
cites work: Symmetric statistics, Poisson point processes, and multiple Wiener integrals
cites work: On the weak limits of elementary symmetric polynomials
cites work: Elementary symmetric polynomials of increasing order
cites work: Large sample theory for U-statistics and tests of fit
cites work: The asymptotic distributions of incomplete U-statistics
cites work: A central limit theorem for generalized quadratic forms
cites work: Q3992833
cites work: A central limit theorem for generalized multilinear forms
cites work: On the Limit Distribution of a Two-Sample von Mises Functional with Variable Kernel
cites work: On Limit Theorems for the Distributions of Generalized von Mises Functionals
cites work: Q3996041
cites work: Random multilinear forms
cites work: Decoupling inequalities for polynomial chaos
cites work: Double stochastic integrals, random quadratic forms and random series in Orlicz spaces
cites work: Q3999009
cites work: Q3662352
cites work: Invariance principle for symmetric statistics
cites work: Decoupling inequalities for multilinear forms in independent symmetric random variables
cites work: Functional limit theorems for random quadratic forms
cites work: Asymptotic behaviour of symmetric polynomial statistics
cites work: Functional limit theorems for U-statistics in the degenerate case
cites work: Q4057863
cites work: A Functional Limit Theorem for Homogeneous U-Statistics with a Degenerate Kernel
cites work: Some Limit Theorems for Polynomials of Second Degree
cites work: Limit Theorems for Multilinear Forms and Quasipolynomial Functions
cites work: On the Distribution of a Quadratic Form in Many Random Variables
cites work: Limit theorems for polylinear forms
cites work: Asymptotic distribution of symmetric statistics
cites work: On the Limit Distributions of Random Symmetric Functions
cites work: On Limit Distributions of Random Symmetric Polynomials
cites work: Approximation Theorems of Mathematical Statistics
cites work: Asymptotic normality of permutation statistics derived from weighted sums of bivariate functions
cites work: Weighted Rank Statistics for Simple Linear Regression
cites work: Q3729765
cites work: Q4136031
description: scientific article; zbMATH DE number 501135