Electronic Communications in Probability

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

J. Armando Dominguez-Molina, Víctor Pérez-Abreu, and Alfonso Rocha-Arteaga

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; Lévy processes 60G57: Random measures

Infinitely divisible random matrix matrix subordinator Bercovici-Pata bijection matrix semimartingale matrix compound Poisson

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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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