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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
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