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

TL;DR

This paper establishes the equivalence between multivariate MCARMA processes and continuous-time state space models, and proves the complete regularity of sampled process innovations under mild conditions, with broad practical applicability.

Contribution

It demonstrates the equivalence of MCARMA processes and state space models, and proves the complete regularity of innovations for sampled processes under general conditions.

Findings

Innovations are exponentially completely regular ($eta$-mixing) under mild assumptions.

Continuity conditions are satisfied in most practical scenarios.

Results apply to various Lévy processes including Gaussian, compound Poisson, and those with infinite Lévy measures.

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.