Characterization of probability distributions by Poincaré-type inequalities
type: Item
instance of: scholarly article
title: Characterization of probability distributions by Poincaré-type inequalities
zbMATH Open document ID: 0612.60013
author: Louis H. Y. Chen
author: J. H. Lou
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77293
review text: The author considers generalizations of the following characterization of the normal distribution, due to \textit{A. A. Borovkov} and \textit{S. A. Utev} [Teor. Veroyatn. Primen. 28, No.2, 209-218 (1983; Zbl 0511.60016); English translation in Theory Probab. Appl. 28, 219-228 (1984)]: \[ (*)\quad U_ X:=\sup_{g}\frac{var g(X)}{\sigma^ 2E(g'(X))^ 2}=1 \] if and only if X has a normal distribution. When X is taken to be m- dimensional and the denominator of \(U_ X\) is generalized such as to include a positive definite \(m\times m\) matrix \(\Sigma\) and a measure \(\mu\), then the analogue of (*) characterizes the infinitely divisible distributions having a normal component with covariance matrix \(\Sigma\) and canonical measure \(\mu\) ; this is the content of Theorem 2.1. Related results include a limit version of Theorem 2.1, more general characterization theorems, and some discrete analogues.
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 3990490
zbMATH Keywords: characterization of the normal distribution
zbMATH Keywords: infinitely divisible distributions
reviewed by: Fred W. Steutel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3990490