A new factorization property of the selfdecomposable probability measures

Abstract

We prove that the convolution of a selfdecomposable distribution with its background driving law is again selfdecomposable if and only if the background driving law is s-selfdecomposable. We will refer to this as the factorization property of a selfdecomposable distribution; let L^f denote the set of all these distributions. The algebraic structure and various characterizations of L^f are studied. Some examples are discussed, the most interesting one being given by the Lévy stochastic area integral. A nested family of subclasses L^f_n, n≥0, (or a filtration) of the class L^f is given.