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 \textit{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 $L_n^f,n\ge 0,$ (or a filtration) of the class $L^f$ is given.