Limit theorems for free L\'evy processes

TL;DR

This paper investigates limit theorems for free and Boolean Lévy processes, demonstrating convergence to free and Boolean stable laws at small or large times, and identifying connections with known distributions.

Contribution

It establishes new convergence results for free and Boolean Lévy processes, including the identification of log free stable laws with index 1 as the Dykema-Haagerup distribution.

Findings

Convergence to log free stable laws for multiplicative free Lévy processes.

Additive free Lévy processes converge to free stable laws.

Limit theorems for Boolean Lévy processes yield log Boolean stable laws.

Abstract

We consider different limit theorems for additive and multiplicative free L\'evy processes. The main results are concerned with positive and unitary multiplicative free L\'evy processes at small time, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large time to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean L\'evy processes at small time, obtaining log Boolean stable laws in the limit.