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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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