New limit theorems related to free multiplicative convolution
Volume 214 / 2013
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.
