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 $(M_{d})_{d\geq1}$ 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 in [6] and [10]. 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 an $\mathbb{C}^{d}$-valued Lévy process. In particular, we have the corresponding result for matrix subordinators with jumps of rank one associated to the random matrix ensembles $(M_{d})_{d\geq1}$