New limit theorems related to free multiplicative convolution

Noriyoshi Sakuma¹ ; Hiroaki Yoshida²

¹Department of Mathematics Aichi University of Education 1 Hirosawa, Igaya-cho, Kariya-shi 448-8542 Japan
²Department of Information Sciences Ochanomizu University 2-1-1, Otsuka, Bunkyo, Tokyo 112-8610 Japan

Studia Mathematica, Tome 214 (2013) no. 3, pp. 251-264

Résumé

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.

Détail

DOI : 10.4064/sm214-3-4

Keywords: boxplus boxtimes uplus additive multiplicative boolean additive convolutions respectively probability measure infty finite second moment scaling limit boxtimes boxplus goes infinity mathcal transform its limit distribution represented lamberts w function deduce limiting distribution freely infinitely divisible lognormal distribution classical similar limit theorem replacing additive convolution boolean convolution