A REPRESENTATION FOR CHARACTERISTIC FUNCTIONALS OF STABLE RANDOM MEASURES WITH VALUES IN SAZONOV SPACES

We deal with a Sazonov space (X: real separable) valued symmetric alpha stable random measure Phi with independent increments on the measurable space (R(k), B(R(k))). A pair (k, mu), called here a control pair, for which k : X x R(k) -> R(+), mu a positive measure on (R(k), B(R(k))), is introduced. It is proved that the law of Phi is governed by a control pair; and every control pair will induce such Phi. Moreover, k is unique for a given mu. Our derivations are based on the Generalized Bochner Theorem and the Radon-Nikodym Theorem for vector measures.