In multivariate stationary time series, the inverse of the spectral density matrix encodes many important properties such as partial correlations, graphical models, and autoregressive representations. This is not the case for nonstationary time series where the relevant information is in the inverse infinite-dimensional covariance matrix operator associated with the multivariate time series. This requires the study of the relationship between the covariance of multivariate nonstationary time series and its inverse. If a row/column of the infinite-dimensional covariance matrix decays at a certain rate, then that percentage (up to the coefficient) migrates to the row/column of the inverse covariance matrix. It is used to obtain the non-stationary autoregressive representation of the time series and the Baxterian bounds between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projections. The preceding results form the basis for the subsequent analysis of the localized time series. In particular, we show that the smoothness property of the covariance matrix is transferred to (i) the inverse covariance, (ii) the parameters of the vector autoregressive representation, and (iii) the partial covariances. All results are set so that the constants of interest depend only on the eigenvalues of the covariance matrix and are applicable in high-dimensional settings with non-divergent eigenvalues.
