Asymptotically Equivalent Sequences of Matrices and Multivariate ARMA Processes

The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary (WSS) processes. A multivariate random process is said to be asymptotically WSS if it has constant mean and the sequence of its autocorrelation matrices is asymptotically equivalent (a.e.) to the sequence of autocorrelation matrices of some multivariate WSS process. It is shown that this class of processes contains meaningful processes other than multivariate WSS processes. In particular, we give sufficient conditions for multivariate moving average (MA) processes, multivariate autoregressive (AR) processes and multivariate autoregressive moving average (ARMA) processes to be asymptotically WSS. Furthermore, in order to solve multiple-input-multiple-output (MIMO) problems in communications and signal processing involving this kind of processes, we extend the Gray definition of a.e. sequences of matrices and his main results on these sequences to non-square matrices. As an example, the derived results on a.e. sequences of non-square matrices are applied to compute the differential entropy rate and the minimum mean square error (MMSE) for a linear predictor of a multivariate asymptotically WSS process.