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 Lf denote the set of all these distributions. The algebraic structure and various characterizations of Lf are studied. Some examples are discussed, the most interesting one being given by the Lévy stochastic area integral. A nested family of subclasses Lfn, n≥0, (or a filtration) of the class Lf is given.
"A new factorization property of the selfdecomposable probability measures." Ann. Probab. 32 (2) 1356 - 1369, April 2004. https://doi.org/10.1214/009117904000000225