  For multivariate stationary time series many important properties, such as
partial correlation, graphical models and autoregressive representations are
encoded in the inverse of its spectral density matrix. This is not true for
nonstationary time series, where the pertinent information lies in the inverse
infinite dimensional covariance matrix operator associated with the
multivariate time series. This necessitates the study of the covariance of a
multivariate nonstationary time series and its relationship to its inverse. We
show that if the rows/columns of the infinite dimensional covariance matrix
decay at a certain rate then the rate (up to a factor) transfers to the
rows/columns of the inverse covariance matrix. This is used to obtain a
nonstationary autoregressive representation of the time series and a
Baxter-type bound between the parameters of the autoregressive infinite
representation and the corresponding finite autoregressive projection. The
aforementioned results lay the foundation for the subsequent analysis of
locally stationary time series. In particular, we show that smoothness
properties on the covariance matrix transfer to (i) the inverse covariance (ii)
the parameters of the vector autoregressive representation and (iii) the
partial covariances. All results are set up in such a way that the constants
involved depend only on the eigenvalue of the covariance matrix and can be
applied in the high-dimensional settings with non-diverging eigenvalues.
