Rational $q\times q$ Carathéodory Functions and Central Non-negative Hermitian Measures

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 $q\times q$ Carathéodory 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.