Operator-stable laws

Sharpe investigated the structure of full operator-stable measures [mu] on a vector group V and obtained decompositions, [mu] = [mu]1 * [mu]2 and V = V1 [circle plus operator] V2, in terms of the Gaussian component [mu]1 and the Poisson component [mu]2. The subspaces V1 and V2 are here identified in terms of an exponent B for [mu]. Sharpe also pointed out that the Lévy measure M of [mu] is a mixture of Lévy measures concentrated on single orbits of tB. Here, an explicit representation is obtained for M as such a mixture by constructing a measure on the unit sphere. Also, necessary and sufficient conditions are given that a Lévy measure be the Lévy measure of a full operator-stable measure. The final result deals with full Gaussian measures [mu] and establishes the connection between its covariance operator and the class of all exponents of [mu].