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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

Robert Stelzer

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.

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

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.

Paper Structure

This paper contains 2 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Proposition 2.1

For the output process $\mathbf{Y}$ of a Lévy-driven state space model of the form (P3.5) there exist a $p\in\mathbb{N}$ and matrices $\mathcal{A,B,C}$ of the form (P3.4) such that $\mathbf{Y}$ is the output process of an MCARMA state space representation (linear state space model in observer canoni

Theorems & Definitions (6)

  • Proposition 2.1: Observer canonical form/MCARMA process representation
  • proof
  • Proposition 2.2: Controller canonical form
  • proof
  • Proposition 2.3: Matrix fraction representations
  • proof