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arXiv · 1103.6156

New limit theorems related to free multiplicative convolution

Noriyoshi Sakuma·Hiroaki Yoshida
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Abstract

Let $\boxplus$, $\boxtimes$ and $\uplus$ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure $μ$ on $[0,\infty)$ with finite second moment, we find the scaling limit of $(μ^{\boxtimes N})^{\boxplus N}$ as $N$ goes to infinity. The $\mathcal{R}$--transform of the limit distribution can be represented by the Lambert's $W$ function. We also find similar limit theorem by replacing the free additive convolution with the boolean convolution.

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Noriyoshi Sakuma, Hiroaki Yoshida. 2013-03-21. New limit theorems related to free multiplicative convolution. https://arxiv.org/abs/1103.6156

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