One-dimensional marginals of operator stable laws and their domains of attraction

Abstract

Operator stable laws are the weak limits of ane normalized partial sums of i.i.d. random vectors. It is known that the one-dimensional marginals of operator stable laws need not be stable, or even attracted to a stable law. In this paper we show that for any operator stable law, there exists a basis in which the marginals along every coordinate axis are attracted to a stable or semistable law. This connection between operator stable and semistable laws is new and surpris- ing. We also characterize those operator stable laws whose marginals are stable or semistable. Finally we consider the marginals of random vectors attracted to some operator stable law.