Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes
Abstract:
The class of multivariate Lévy-driven auto-regressive moving-average (MCARMA) processes, the continuous-time
analogues 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 (β-mixing) under a mild continuity assumption on
the driving Lévy process. It is verified that this continuity assumption is satisfied in most practically relevant situations
including the case when the driving Lévy process has a non-singular Gaussian component, is compound Poisson with
an absolutely continuous jump size distribution or has an infinite Lévy measure admitting a density around zero. «
The class of multivariate Lévy-driven auto-regressive moving-average (MCARMA) processes, the continuous-time
analogues 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 (β-mixing) under a mild continuity assumption on
the driving Lévy process. It is verified that t... »
Keywords:
complete regularity, linear innovations, multivariate CARMA process, sampling, state space representation, strong mixing, vector ARMA process.