Open Access
April 2004 A new factorization property of the selfdecomposable probability measures
Aleksander M. Iksanov, Zbigniew J. Jurek, Bertram M. Schreiber
Ann. Probab. 32(2): 1356-1369 (April 2004). DOI: 10.1214/009117904000000225


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.


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Aleksander M. Iksanov. Zbigniew J. Jurek. Bertram M. Schreiber. "A new factorization property of the selfdecomposable probability measures." Ann. Probab. 32 (2) 1356 - 1369, April 2004.


Published: April 2004
First available in Project Euclid: 18 May 2004

zbMATH: 1046.60002
MathSciNet: MR2060300
Digital Object Identifier: 10.1214/009117904000000225

Primary: 60B12 , 60E07
Secondary: 60G51 , 60H05

Keywords: background driving Lévy process , class L , class U , factorization property , Infinitely divisible , Lévy exponent , Lévy spectral measure , Lévy stochastic area integral , Selfdecomposable , s-selfdecomposable , Stable

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 2 • April 2004
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