New limit theorems related to free multiplicative convolution

TL;DR

This paper establishes new limit theorems for free multiplicative convolution, revealing the asymptotic behavior of scaled convolutions and expressing the limit's transform via Lambert's W function, with parallels in boolean convolution.

Contribution

It introduces novel limit theorems for free multiplicative convolution and provides explicit transform representations involving Lambert's W function.

Findings

The limit of $(er{ ext{measure}}^{oxtimes N})^{oxplus N}$ exists as $N$ approaches infinity.

The limit distribution's $al R$-transform involves Lambert's W function.

Similar limit results are obtained for boolean convolution.

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 the scaling limit of $(\mu^{\boxtimes N})^{\boxplus N}$ 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.