STATISTICS& PROBABILITY LLrl'I'ERS Symmetry groups in d-space

We show that any finite-dimensional compact Lie group is isomorphic to the symmetry group of a full probability measure. The novelty of our proof is that an explicit formula for the measure and its support is given in terms of the Lie group. We also construct a full operator stable probability measure whose symmetry group has as its tangent space the tangent space of a given group. This provides a method for constructing an operator stable probability measure having a specified collection of exponents. A characterization f the compact groups of operators on a finite-dimensional space which can be the symmetry group of a full probability measure on that same space is given