Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution

Victor Pérez-Abreu () and Noriyoshi Sakuma ()
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Victor Pérez-Abreu: CIMAT
Noriyoshi Sakuma: Keio University

Journal of Theoretical Probability, 2012, vol. 25, issue 1, 100-121

Abstract: Abstract Let I * and I ⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici–Pata bijection between I * and I ⊞ . The class type W of symmetric distributions in I ⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ⊞ and the free counterpart under Λ of the positive distributions in I * is established. It is shown that the class type W does not include all symmetric distributions in I ⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I *. Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.

Keywords: Free convolutions; Type G law; Free stable law; Free compound distribution; Bercovici–Pata bijection; 46L54; 15A52 (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-010-0288-5

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