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 $(M_d{)}_{d\ge 1}$ 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 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 $(M_d{)}_{d\ge 1}$

13 pages

References in corpus (3)

• Multivariate supOU processes
• On the definition, stationary distribution and second order structure of positive semidefinite Ornstein--Uhlenbeck type processes
• Multivariate COGARCH(1,1) processes