Representation of certain infinitely divisible probability measures on Banach spaces

Abstract

Subclasses L 0 ⊃ L 1 ⊃ … ⊃ L ∞ of the class L 0 of self-decomposable probability measures on a Banach space are defined by means of certain stability conditions. Each of these classes is closed under translation, convolution and passage to weak limits. These subclasses are analogous to those defined earlier by K. Urbanik on the real line and studied in that context by him and by the authors. A representation is given for the characteristic functionals of the measures in each of these classes on conjugate Banach spaces. On a Hilbert space it is shown that L ∞ is the smallest subclass of L 0 with the closure properties above containing all the stable measures.