## Electronic Communications in Probability

### Covariation representations for Hermitian Lévy process ensembles of free infinitely divisible distributions

#### Abstract

It is known that the so-called Bercovici-Pata bijection can be explained in terms of certain Hermitian random matrix ensembles (M<sub>d</sub>)<sub>d≥1</sub> whose asymptotic spectral distributions are free infinitely divisible. We investigate Hermitian Lévy processes with jumps of rank one associated to these random matrix ensembles introduced by Benaych-Georges and Cabanal-Duvillard. A sample path approximation by covariation processes for these matrix Lévy processes is obtained. As a general result we prove that any d×d complex matrix subordinator with jumps of rank one is the quadratic variation of a $\mathbb{C}^d$-valued Lévy process. In particular, we have the corresponding result for matrix subordinators with jumps of rank one associated to the random matrix ensembles (M<sub>d</sub>)<sub>d≥1</sub>.<br />

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 6, 14 pp.

Dates
Accepted: 17 January 2013
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465315545

Digital Object Identifier
doi:10.1214/ECP.v18-2113

Mathematical Reviews number (MathSciNet)
MR3019669

Zentralblatt MATH identifier
1308.60056

Rights

#### Citation

Dominguez-Molina, J. Armando; Pérez-Abreu, Víctor; Rocha-Arteaga, Alfonso. Covariation representations for Hermitian Lévy process ensembles of free infinitely divisible distributions. Electron. Commun. Probab. 18 (2013), paper no. 6, 14 pp. doi:10.1214/ECP.v18-2113. https://projecteuclid.org/euclid.ecp/1465315545

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