Electronic Journal of Probability

Limit theorems for free Lévy processes

Octavio Arizmendi and Takahiro Hasebe

Full-text: Open access

Abstract

We consider different limit theorems for additive and multiplicative free Lévy processes. The main results are concerned with positive and unitary multiplicative free Lévy processes at small times, 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 times 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 Lévy processes at small times, obtaining log Boolean stable laws in the limit.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 101, 36 pp.

Dates
Received: 29 November 2017
Accepted: 14 September 2018
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1538618571

Digital Object Identifier
doi:10.1214/18-EJP224

Subjects
Primary: 46L54: Free probability and free operator algebras
Secondary: 46L53: Noncommutative probability and statistics 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems

Keywords
free Lévy processes multiplicative convolutions Boolean independence limit theorems, small times

Rights
Creative Commons Attribution 4.0 International License.

Citation

Arizmendi, Octavio; Hasebe, Takahiro. Limit theorems for free Lévy processes. Electron. J. Probab. 23 (2018), paper no. 101, 36 pp. doi:10.1214/18-EJP224. https://projecteuclid.org/euclid.ejp/1538618571


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