Necessary and sufficient conditions for AR vector processes to be stationary: Applications in information theory and in statistical signal processing

Abstract

As the correlation matrices of stationary vector processes are block Toeplitz, autoregressive (AR) vector processes are non-stationary. However, in the literature, an AR vector process of finite order is said to be “stationary” if it satisfies the so-called stationarity condition (i.e., if the spectral radius of the associated companion matrix is less than one). Since the term “stationary” is used for such an AR vector process, its correlation matrices should “somehow approach” the correlation matrices of a stationary vector process, but the meaning of “somehow approach” has not been mathematically established in the literature. In the present paper we give necessary and sufficient conditions for AR vector processes to be “stationary”. These conditions show in which sense the correlation matrices of an AR “stationary” vector process asymptotically behave like block Toeplitz matrices. Applications in information theory and in statistical signal processing of these necessary and sufficient conditions are also given.