Barndorff-Nielsen, Ole Eiler3; Thorbjørnsen, Steen3
1 Department of Mathematical Sciences, Faculty of Science, Aarhus University, Aarhus University2 Department of Mathematics, Science and Technology, Aarhus University3 Department of Mathematics, Science and Technology, Aarhus University
In this paper we introduce and study a regularizing one-to-one mapping from the class of one-dimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjørnsen, A connection between free and classical infinite divisibility, Inf. Dim. Anal. Quant. Probab. 7 (2004) 573–590], where we introduced a one-to-one mapping from the class of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of in the present paper, we deduce further properties of . In particular it is proved that maps the class of selfdecomposable laws onto the so called Thorin class . Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family of one-to-one mappings , which interpolates smoothly between ( α=0 ) and the identity mapping on ( α=1 ). We prove that each of the mappings shares many of the properties of . In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.
Stochastic Processes and Their Applications, 2006, Vol 116, Issue 3, p. 423-446