A+ CATEGORY SCIENTIFIC UNIT

New limit theorems related to free multiplicative convolution

Volume 214 / 2013

Noriyoshi Sakuma, Hiroaki Yoshida Studia Mathematica 214 (2013), 251-264 MSC: Primary 46L54; Secondary 15A52. DOI: 10.4064/sm214-3-4

Abstract

Let $\boxplus $, $\boxtimes $, and $\uplus $ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure $\mu $ on $[0,\infty )$ with finite second moment, we find a scaling limit of $(\mu ^{\boxtimes N})^{\boxplus N}$ as $N$ goes to infinity. The $\mathcal {R}$-transform of its limit distribution can be represented by Lambert's $W$-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free additive convolution with boolean convolution.

Authors

  • Noriyoshi SakumaDepartment of Mathematics
    Aichi University of Education
    1 Hirosawa, Igaya-cho, Kariya-shi
    448-8542 Japan
    e-mail
  • Hiroaki YoshidaDepartment of Information Sciences
    Ochanomizu University
    2-1-1, Otsuka, Bunkyo, Tokyo
    112-8610 Japan
    e-mail

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