On convergence and convolutions of random signed measures
type: Item
instance of: scholarly article
title: On convergence and convolutions of random signed measures
zbMATH Open document ID: 1175.60030
author: Richard Nickl
published in: Journal of Theoretical Probability
publication date: 2009-04-24T00:00:00Z
review text: Let \(\mu_n\) be a sequence of random signed measures on the group \(G\) equal to either \(\mathbb{R}^d\) or the \(d\)-dimensional Torus \(T^d\). There are proved two general theorems stating that if \(\mu_n\) converges in some weak sense and a function \(K\) on \(G\) belongs to an appropriately chosen Sobolev or Besov space then the convolution \(\mu_n*K\) converges in law, in probability or almost surely in the Banach spaces \(C_0(G)\) or \(L^p(G)\). Using them convergence of distributional derivatives of \(\mu_n*K\) is also established. Examples for \(\mu_n\) covered are empirical processes (possibly arising from dependent data) and random signed measures defined by nonparametric kernel or wavelet based density estimators. An application of the general theorems to convolutions of density estimators is studied in detail. In particular, recent results of \textit{A. Schick} and \textit{W. Wefelmeyer} [J. Nonparametric Stat. 16, No. 6, 925--935 (2004; Zbl 1062.62065)] and \textit{E. Giné} and \textit{D. M. Mason} [Ann. Stat. 35, No. 3, 1105--1145 (2007; Zbl 1175.60017)] are extended and put into a general scheme.
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 62G07
zbMATH DE Number: 5547309
zbMATH Keywords: random signed measure
zbMATH Keywords: convolution
zbMATH Keywords: Sobolev space
zbMATH Keywords: limit theorems
zbMATH Keywords: empirical probess
zbMATH Keywords: density estimator
zbMATH Keywords: Banach space
MaRDI profile type: Publication
full work available at URL: s10959-008-0177-3
OpenAlex ID: W2140942254
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DOI: 10.1007/S10959-008-0177-3
description: scientific article; zbMATH DE number 5547309