Rates of weak convergence for images of measures by families of mappings
type: Item
instance of: scholarly article
title: Rates of weak convergence for images of measures by families of mappings
zbMATH Open document ID: 1005.60013
DOI: 10.1016/S0167-7152(01)00038-4
author: André Mas
published in: Statistics \& Probability Letters
publication date: 2002-09-05T00:00:00Z
review text: Let \(E_1\) be a Banach space and let \(\mu_n\) and \(\mu\) be Borel measures on \(E_1\) such that \(\mu_n\to \mu\) weakly. Given a certain set \(\mathcal F\) of continuous functions from \(E_1\) into another Banach space \(E_2\), for each \(f\in \mathcal F\) it follows that \(\mu_n\circ f^{-1}\to \mu\circ f^{-1}\) weakly in \(E_2\) as well. An interesting question is as follows: Compare for \(f\in\mathcal F\) the speed of convergence (w.r.t. the Prokhorov distance \(\pi\)) of \(\mu_n\circ f^{-1}\) to \(\mu\circ f^{-1}\) with that of \(\pi(\mu_n,\mu)\) in dependence of properties of \(\mathcal F\). As shown by \textit{W. Whitt} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 29, 39-44 (1974; Zbl 0268.60004)] the speed of convergence of the image measures is (up to a constant) the same for \(\mathcal F\)'s bounded w.r.t.~the Hölder norm. The present paper treats the unbounded case, yet with some control about the Hölder continuity of functions \(f\) belonging to \(\mathcal F\).
reviewed by: Werner Linde
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 46E27
zbMATH DE Number: 1796652
zbMATH Keywords: weak convergence
zbMATH Keywords: Banach spaces
zbMATH Keywords: Prokhorov metric
MaRDI profile type: Publication
cites work: Q5560061
cites work: Q4219536
cites work: Distances of Probability Measures and Random Variables
cites work: Q3471245
cites work: Preservation of rates of convergence under mappings
description: scientific article; zbMATH DE number 1796652