Factorization and discrete-time representation of multivariate CARMA processes

TL;DR

This paper demonstrates that multivariate MCARMA processes can be represented as sums of complex Ornstein-Uhlenbeck processes, providing new insights into their autocovariance structure and discrete-time sampling properties.

Contribution

It introduces a novel representation of MCARMA processes as sums of Ornstein-Uhlenbeck processes and links continuous-time models to discrete-time VARMA processes, enhancing understanding of their autocovariance.

Findings

Representation of MCARMA as sums of Ornstein-Uhlenbeck processes

Discrete-time MCARMA sampled processes are weak VARMA(p,p-1)

Factorization of autocovariance function for statistical inference

Abstract

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.