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April 2002 Self-decomposability and Lévy processes in free probability
Ole E. Barndorff-Nielsen, Steen Thorbjørnsen
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Bernoulli 8(3): 323-366 (April 2002).


In this paper we study the bijection, introduced by Bercovici and Pata, between the classes of infinitely divisible probability measures in classical and in free probability. We prove certain algebraic and topological properties of that bijection (in the present paper denoted Λ), and those properties are then used to show, in particular, that Λ maps the class of classically self-decomposable probability measures onto the natural free counterpart which we define here. Further, we study Lévy processes in free probability and use the properties of Λ to construct stochastic integrals with respect to such processes. In particular, we derive the free analogue of the integral representation of self-decomposable random variables.


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Ole E. Barndorff-Nielsen. Steen Thorbjørnsen. "Self-decomposability and Lévy processes in free probability." Bernoulli 8 (3) 323 - 366, April 2002.


Published: April 2002
First available in Project Euclid: 8 March 2004

zbMATH: 1024.60022
MathSciNet: MR2003C:60031

Keywords: Free additive convolution , free and classical infinite divisibility , free Lévy processes , free Ornstein-Uhlenbeck processes , free self-decom\-posability , free stochastic integrals

Rights: Copyright © 2002 Bernoulli Society for Mathematical Statistics and Probability

Vol.8 • No. 3 • April 2002
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