## A new factorization property of the selfdecomposable probability measures.(English)Zbl 1046.60002

Summary: 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 $$L^f_n$$, $$n\geq 0$$, (or a filtration) of the class $$L^f$$ is given.

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60E07 Infinitely divisible distributions; stable distributions 60G51 Processes with independent increments; Lévy processes 60H05 Stochastic integrals
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### References:

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