Electronic Journal of Probability

Limit theorems for free Lévy processes

Octavio Arizmendi and Takahiro Hasebe

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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

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

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

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Zentralblatt MATH identifier

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

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

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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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