Rational $q\times q$ Carath\'eodory Functions and Central Non-negative Hermitian Measures

TL;DR

This paper provides explicit representations of central measures and Carathéodory functions for matrix-valued sequences, linking them to spectral measures of multivariate autoregressive processes.

Contribution

It introduces explicit formulas connecting central measures, Carathéodory functions, and spectral measures in the context of matrix-valued sequences.

Findings

Explicit representation of central measures for Toeplitz matrix sequences

Connection between Carathéodory functions and spectral measures

Representation of spectral measure in terms of covariance sequence

Abstract

We give an explicit representation of central measures corresponding to finite Toeplitz non-negative definite sequences of complex $q\times q$ matrices. Such measures are intimately connected to central $q\times q$ Carath\'eodory functions. This enables us to prove an explicit representation of the non-stochastic spectral measure of an arbitrary multivariate autoregressive stationary sequence in terms of the covariance sequence.