The Annals of Probability

Elliptical Symmetry and Characterization of Operator-Stable and Operator Semi-stable Measures

A. Luczak

Abstract

The problem of the elliptical symmetry of an operator-stable measure on a finite dimensional vector space was studied by J. P. Holmes, W. N. Hudson and J. D. Mason (1982). The aim of this paper is to consider an analogous question for operator semi-stable measures. We prove a theorem characterizing an elliptically symmetric full operator semi-stable measure. After such a generalization some results for operator-stable measures are obtained as corollaries. At the same time, the methods of proofs seem to be simpler than those in [1], in particular, the theory of Lie algebras is not involved. We also give a description of a full operator-stable and operator semi-stable measure $\mu$ in terms of its quasi-decomposability group, namely the group $\mathbb{G}(\mu) = \{t > 0: \exists A \in \operatorname{Aut} V, h \in V \quad\text{such that}\quad\mu^t = A\mu \ast \delta(h)\}.$

Article information

Source
Ann. Probab., Volume 12, Number 4 (1984), 1217-1223.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993151

Digital Object Identifier
doi:10.1214/aop/1176993151

Mathematical Reviews number (MathSciNet)
MR757779

Zentralblatt MATH identifier
0549.60007

JSTOR