On a characterization theorem for a Cauchy distribution on Abelian groups
type: Item
instance of: scholarly article
title: On a characterization theorem for a Cauchy distribution on Abelian groups
zbMATH Open document ID: 0856.60009
DOI: 10.1007/BF02113933
published in: Siberian Mathematical Journal
publication date: 1997-02-17T00:00:00Z
review text: This paper generalizes the following classical characterization of Cauchy distributions on \(\mathbb{R}\) to second countable, locally compact Abelian groups \(G\): If \(X_1,\dots, X_n\) are i.i.d. random variables with distribution \(\mu\) such that \(nX_1\) and \(X_1+\cdots+ X_n\) are identically distributed for two values \(n_1\), \(n_2\) of \(n\) with \(\ln n_1/\ln n_2\not\in \mathbb{Q}\), then \(\mu\) is a Cauchy distribution. For the generalization, \(\mu\in M^1(G)\) is called a Cauchy distribution if \(\widehat \mu(y)= y(x)\cdot \exp(- \sqrt{\varphi(y)})\) for some \(x\in G\) and some nonnegative quadratic form \(\varphi\) on the dual group \(\widehat G\). It is proved that, up to a technical additional term concerning Haar measures of compact subgroups of \(G\), the characterization above remains valid if and only if the connected component \(C\) of the identity in \(G\) has \(\dim C\leq 1\), and if there is no element in \(C\) of order \(p\) for some prime \(p\).
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 22B05
zbMATH DE Number: 923492
zbMATH Keywords: locally compact Abelian groups
zbMATH Keywords: Cauchy distribution
zbMATH Keywords: compact subgroups
zbMATH Keywords: characterization
author: G. M. Fel'dman
reviewed by: Michael Voit
MaRDI profile type: Publication
cites work: Q4843641
cites work: Characterizing the Gaussian distribution on groups by the uniform distribution of a monomial and linear statistics
cites work: Groups admitting a characterization of the Gaussian distribution by the equidistribution of a monomial and linear statistics
cites work: Probability distributions on locally compact Abelian groups
cites work: Q3308761
full work available at URL: https://doi.org/10.1007/bf02113933
OpenAlex ID: W2081793491
description: scientific article; zbMATH DE number 923492