Remarks on characterization of normal and stable distributions
type: Item
instance of: scholarly article
title: Remarks on characterization of normal and stable distributions
zbMATH Open document ID: 0776.60017
DOI: 10.1007/BF01047582
published in: Journal of Theoretical Probability
publication date: 1993-11-24T00:00:00Z
review text: Existing results on the characterization of stable distributions lead one to expect that \[ E| tX+uY|^ p=A(| t|^ q+| u|^ q)^{p/q}\quad\text{for all } t,u\in\mathbb{R} \] with some \(A>0\), \(X\) and \(Y\) independent and identical in distribution, implies that \(X\) has a stable distribution. The author showed in a previous paper that this does not hold and now continues to show that even under additional restrictions on the behaviour of \(P(| X|\geq x)\), \(x\to\infty\), such a characterization is not valid. It is shown that, if the above equation holds for \(q=2\) and two different positive values of \(p\neq 2m\) for integer \(m\), then \(X\) has a normal distribution.
reviewed by: Rudolf Grübel
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 62E10
zbMATH DE Number: 179072
zbMATH Keywords: characterization of stable distributions
zbMATH Keywords: normal distribution
author: Michael Braverman
MaRDI profile type: Publication
cites work: The Characteristic Properties of Normal and Stable Distributions
cites work: On a Way of Characterizing Probability Distributions
cites work: A characterization of probability distributions by mean values
cites work: Remarks on properties of probability distributions determined by conditional moments
cites work: Q5598785
cites work: Q3673748
description: scientific article; zbMATH DE number 179072