# zbMATH — the first resource for mathematics

Covariation representations for Hermitian Lévy process ensembles of free infinitely divisible distributions. (English) Zbl 1308.60056
Summary: It is known that the so-called Bercovici-Pata bijection can be explained in terms of certain Hermitian random matrix ensembles $$(M_{d})_{d\geq 1}$$ 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 $$d\times d$$ 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 $$(M_{d})_{d\geq 1}$$.

##### MSC:
 60G51 Processes with independent increments; Lévy processes 60B20 Random matrices (probabilistic aspects) 60E07 Infinitely divisible distributions; stable distributions 60G57 Random measures
Full Text: