New limit theorems related to free multiplicative convolution

Noriyoshi Sakuma, and Hiroaki Yoshida. "New limit theorems related to free multiplicative convolution." Studia Mathematica 214.3 (2013): 251-264. <http://eudml.org/doc/285438>.

@article{NoriyoshiSakuma2013,
abstract = {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 $(μ^\{⊠ N\})^\{⊞ N\}$ 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.},
author = {Noriyoshi Sakuma, Hiroaki Yoshida},
journal = {Studia Mathematica},
keywords = {free convolutions; Boolean convolution; infinite divisibility; Lambert’s -function; limit theorem},
language = {eng},
number = {3},
pages = {251-264},
title = {New limit theorems related to free multiplicative convolution},
url = {http://eudml.org/doc/285438},
volume = {214},
year = {2013},
}

TY - JOUR
AU - Noriyoshi Sakuma
AU - Hiroaki Yoshida
TI - New limit theorems related to free multiplicative convolution
JO - Studia Mathematica
PY - 2013
VL - 214
IS - 3
SP - 251
EP - 264
AB - 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 $(μ^{⊠ N})^{⊞ N}$ 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.
LA - eng
KW - free convolutions; Boolean convolution; infinite divisibility; Lambert’s -function; limit theorem
UR - http://eudml.org/doc/285438
ER -