Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of as N goes to infinity. The -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-4,
author = {Noriyoshi Sakuma and Hiroaki Yoshida},
title = {New limit theorems related to free multiplicative convolution},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {251-264},
zbl = {1280.46044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-4}
}
Noriyoshi Sakuma; Hiroaki Yoshida. New limit theorems related to free multiplicative convolution. Studia Mathematica, Tome 215 (2013) pp. 251-264. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-4/