Stochastic Integral and Covariation Representations for Rectangular Lévy Process Ensembles

A Bercovici-Pata bijection \(\Lambda _{c}\) from the set of symmetric infinitely divisible distributions to the set of ⊞  _c -free infinitely divisible distributions, for certain free convolution ⊞  _c is introduced in Benaych-Georges (Random matrices, related convolutions. Probab Theory Relat Fields 144:471–515, 2009. Revised version of F. Benaych-Georges: Random matrices, related convolutions. arXiv, 2005). This bijection is explained in terms of complex rectangular matrix ensembles whose singular distributions are ⊞  _c -free infinitely divisible. We investigate the rectangular matrix Lévy processes with jumps of rank one associated to these rectangular matrix ensembles. First as general result, a sample path representation by covariation processes for rectangular matrix Lévy processes of rank one jumps is obtained. Second, rectangular matrix ensembles for ⊞  _c -free infinitely divisible distributions are built consisting of matrix stochastic integrals when the corresponding symmetric infinitely divisible distributions under \(\Lambda _{c}\) admit stochastic integral representations. These models are realizations of stochastic integrals of nonrandom functions with respect to rectangular matrix Lévy processes. In particular, any ⊞  _c -free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type \(\int _{0}^{\infty }e^{-t}\mathrm{d}\Psi (t)\), where \(\left \{\Psi (t): t \geq 0\right \}\) is a rectangular matrix Lévy process.