New limit theorems related to free multiplicative convolution

Abstract

Let $⊞$, $⊠$ 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 $({\mu }^{⊠N}{)}^{⊞N}$ 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.

15pages and 2 figures, To appear in Studia Math

References in corpus (2)

• Local limit theorems in free probability theory
• Stieltjes, Poisson and other integral representations for functions of Lambert $W$

Cited by in corpus (1)

• Products of free random variables and k-divisible partitions