  In this paper we show that stationary and non-stationary multivariate
continuous-time ARMA (MCARMA) processes have the representation as a sum of
multivariate complex-valued Ornstein-Uhlenbeck processes under some mild
assumptions. The proof benefits from properties of rational matrix polynomials.
A conclusion is an alternative description of the autocovariance function of a
stationary MCARMA process. Moreover, that representation is used to show that
the discrete-time sampled MCARMA(p,q) process is a weak VARMA(p,p-1) process if
second moments exist. That result complements the weak VARMA(p,p-1)
representation derived in Chambers and Thornton (2012). In particular, it
relates the right solvents of the autoregressive polynomial of the MCARMA
process to the right solvents of the autoregressive polynomial of the VARMA
process; in the one-dimensional case the right solvents are the zeros of the
autoregressive polynomial. Finally, a factorization of the sample
autocovariance function of the noise sequence is presented which is useful for
statistical inference.
