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 , \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.