Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

Abstract

The class of multivariate L\'{e}vy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular ($\beta$-mixing) under a mild continuity assumption on the driving L\'{e}vy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving L\'{e}vy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite L\'{e}vy measure admitting a density around zero.