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

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 $\mathbb{G}(\mu) = \{t > 0: \exists A \in \operatorname{Aut} V, h \in V \quad\text{such that}\quad\mu^t = A\mu \ast \delta(h)\}.$