Multidimensional second-order processes with time-dependent spectra: evolutionary and periodically correlated processes

In this article, we study the multidimensional second-order processes with time-dependent spectra. This is indeed a large class of second-order processes that includes periodically correlated processes and evolutionary processes. We provide new characterization to read an evolutionary process as the Fourier transform of a stationary kernel. We present the corresponding spectral characterization and the moving average representation for doing prediction. The multivariate evolutionary ARMA processes are introduced as well. Insights to the multivariate evolutionary AR(1) processes are provided.