\(\epsilon\)-dependence of \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\)
type: Item
instance of: scholarly article
title: \(\epsilon\)-dependence of \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\)
zbMATH Open document ID: 0607.62006
DOI: 10.1007/BF00966741
published in: Lithuanian Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: According to a famous theorem of Bernstein; if \(X_ 1\) and \(X_ 2\) are independent, then \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\) are independent if and only if \(X_ 1\) and \(X_ 2\) are normal random variables. In this paper the following interesting extension is given. Define the weak metric of the r.v.'s namely \[ \lambda (X,Y)=\lambda (f_ X,f_ Y)=\inf \{\max \{\nu (X,Y;t),t\}:t>0\}, \] where \[ \nu (X,Y;t)=(1/2)\max \{| f_ X(u)-f_ Y(u)|:| u| <1/t\},\quad f_ X(u)=E \exp \{i(u,X)\}. \] Call X and Y (\(\lambda\),\(\epsilon)\)-independent if \(\lambda (f_{(X,Y)},f_ X\cdot f_ Y)\leq \epsilon.\) Theorem: If \(X_ 1\) and \(X_ 2\) are independent and \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\) are (\(\lambda\),\(\epsilon)\)-independent then \[ (1)\quad \inf_{\eta_ j\in \Phi}\lambda (X_ j,\eta_ j)\leq C\cdot \epsilon \quad j=1,2 \] where C is an absolute constant and \(\Phi\) is the class of normal random variables. Moreover the order in (1) is exact.
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60E99
zbMATH DE Number: 3982248
zbMATH Keywords: stability
zbMATH Keywords: characterization of the normal distribution
zbMATH Keywords: independence
zbMATH Keywords: theorem of Bernstein
zbMATH Keywords: weak metric
author: L. B. Klebanov
author: Romanas Yanushkevichius
reviewed by: Antónia Földes
MaRDI profile type: Publication
cites work: On the Robustness of Some Characterizations of the Normal Distribution
cites work: Q3269537
cites work: Stability of solutions of functional equations connected with problems of characterizing probability distributions
description: scientific article; zbMATH DE number 3982248