Modeling and estimation of multivariate discrete and continuous time stationary processes

TL;DR

This paper introduces a comprehensive AR(1)-type characterization for multivariate stationary processes, deriving estimators via Riccati equations, and establishes their consistency and limiting distribution, bridging discrete and continuous time models.

Contribution

It provides a unified AR(1)-type framework for all multivariate stationary processes and develops new estimators with proven consistency and distributional properties.

Findings

Derived continuous time algebraic Riccati equations for parameter estimation.

Proved estimator consistency based on autocovariances.

Established the limiting distribution as a linear function of autocovariance limits.

Abstract

In this paper, we give a AR$(1)$ type of characterization covering all multivariate strictly stationary processes indexed by the set of integers. Consequently, we derive continuous time algebraic Riccati equations for the parameter matrix of the characterization providing us with a natural way to define the corresponding estimator under the assumption of square integrability. In addition, we show that the estimator inherits consistency from autocovariances of the stationary process and furthermore, the limiting distribution is given by a linear function of the limiting distribution of the autocovariances. We also present the corresponding existing results of the continuous time setting paralleling them to the discrete case.