The Annals of Probability

A new factorization property of the selfdecomposable probability measures

Aleksander M. Iksanov, Zbigniew J. Jurek, and Bertram M. Schreiber

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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, n0, (or a filtration) of the class Lf is given.

Article information

Ann. Probab., Volume 32, Number 2 (2004), 1356-1369.

First available in Project Euclid: 18 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Selfdecomposable s-selfdecomposable background driving Lévy process class U class L factorization property infinitely divisible stable Lévy spectral measure Lévy exponent Lévy stochastic area integral


Iksanov, Aleksander M.; Jurek, Zbigniew J.; Schreiber, Bertram M. A new factorization property of the selfdecomposable probability measures. Ann. Probab. 32 (2004), no. 2, 1356--1369. doi:10.1214/009117904000000225.

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