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Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution

Author

Listed:
  • Victor Pérez-Abreu

    (CIMAT)

  • Noriyoshi Sakuma

    (Keio University)

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.

Suggested Citation

  • Victor Pérez-Abreu & Noriyoshi Sakuma, 2012. "Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution," Journal of Theoretical Probability, Springer, vol. 25(1), pages 100-121, March.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:1:d:10.1007_s10959-010-0288-5
    DOI: 10.1007/s10959-010-0288-5
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    Cited by:

    1. A. Kuznetsov, 2020. "On Free Regular and Bondesson Convolution Semigroups," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1493-1505, September.

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