The central limit theorem for \(U\)-processes indexed by Hölder's functions
type: Item
instance of: scholarly article
title: The central limit theorem for \(U\)-processes indexed by Hölder's functions
zbMATH Open document ID: 0808.60030
DOI: 10.1016/0167-7152(94)90234-8
author: Miguel A. Arcones
published in: Statistics \& Probability Letters
publication date: 1995-03-15T00:00:00Z
review text: Given a probability \(P\) on \(R\), a class of real functions \(\mathcal F\) and an i.i.d. sequence of r.v.s \(\{X_ i\}\) with law \(P\), \(\mathcal F\) is said to be \(P\)-Donsker if \(\{n^{1/2} \sum^ n_ 1 (f(X_ i - Pf))\}_{f \in {\mathcal F}}\) converges weakly to a Radon limit as a random element of \(\ell^ \infty({\mathcal F})\). \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 14, 1329-1338 (1986; Zbl 0611.60029)] showed that \({\mathcal F}_ 1\), the class of Hölder's functions of order one, is \(P\)-Donsker iff \(\sum^ \infty_{k = 1} (P(k - 1 < | x | \leq k))^{1/2} < \infty\). The author extends this result for \(1/2 < \alpha \leq 1\) and a positive integer \(m\), to \(\{n^{1/2} (U^ m_ n (f) - Ef\}_{{\mathcal F}_ \alpha}\), which is a generalization of empirical processes, known as \(U\)-processes. For canonical \(U\)-processes there is no similar result. In fact, the author proves that if \(P\) is equal to the Lebesgue measure on \([0,1]\) and \(m \geq 2\), \(\{n^{m/2} U^ m_ n (\pi_{m,m} f) : f \in {\mathcal F}_ 1 ([0,1]^ m)\}\) does not converge to a Radon limit.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 633664
zbMATH Keywords: empirical processes
zbMATH Keywords: \(U\)-statistics
zbMATH Keywords: \(U\)-processes
reviewed by: Adhir K. Basu
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)90234-8
OpenAlex ID: W1998046675
cites work: Limits of canonical \(U\)-processes and \(B\)-valued \(U\)-statistics
cites work: Limit theorems for \(U\)-processes
cites work: The Speed of Mean Glivenko-Cantelli Convergence
cites work: Q3217346
cites work: Some limit theorems for empirical processes (with discussion)
cites work: Empirical processes indexed by Lipschitz functions
cites work: A Class of Statistics with Asymptotically Normal Distribution
cites work: A characterization of maximal ideals in commutative Banach algebras
cites work: Asymptotic distribution of symmetric statistics
cites work: Q5592735
Recommended article: New Donsker classes
Recommended article: Uniform Donsker classes of functions
Recommended article: Uniform Central Limit Theorems
Recommended article: Q3802305
Recommended article: Q4415404
Recommended article: Sum of uniformly distributed random variables and a family of nonalyticC ∞-functions
Recommended article: Q4807518
Recommended article: Q4524737
description: scientific article; zbMATH DE number 633664