Representation theorems for finite predictors of multivariate stationary processes

For a multivariate stationary time series, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF), in terms of the Fourier coefficients of the phase function. The derivation is based on the forward and backward innovations corresponding to the infinite past and future, and the use of a novel alternating projection lemma. The representations are ideal for studying the rates of convergence of the finite prediction error covariances and the PACF as well as that of the finite predictor coefficients through Baxter's inequality. Such results are established for multivariate FARIMA processes with a common fractional differencing order. Even when specialized to a univariate processes, our method and results are more direct and sharper than the known results.