Non-full-rank causal approximations of full-rank multivariate stationary processes with rational spectrum

Abstract

Let Y = ( Y n ) n ϵ ζ be a q-variate wide-sense stationary process with full-rank rational spectral density. We characterize the q-variate stationary processes ( Y ̂ n) n ϵ ζ , causally subordinated to Y, whose spectral densities have rank p < q almost everywhere, and for which E ∥ Y n − Y ̂ n ∥ 2 is minimum.