Correction to: Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes, Bernoulli 18, pp. 46-63, 2012

TL;DR

This paper corrects a flaw in previous proofs regarding the equivalence of MCARMA processes and state space models, establishing their canonical representations and properties of the transfer function.

Contribution

It provides a corrected proof showing that MCARMA processes and linear state space models have both controller and observer canonical forms, clarifying their transfer function representations.

Findings

Corrected proof of equivalence between MCARMA and state space models

Established existence of both controller and observer canonical forms

Confirmed transfer function has both left and right matrix fraction representations

Abstract

A serious flaw in the proof of the equivalence of continuous time state space models and MCARMA processes spotted in Fasen and Schenk (2024) is corrected. We point out that likewise an issue in the proof of Theorem 3.2 in Brockwell and Schlemm (2013) can be resolved and, hence, any MCARMA process and linear state space model has both a controller and an observer canonical representation. Equivalently, the transfer function has both a left and right matrix fraction representation.