Convergence rates for inverse Toeplitz matrix forms

Given a p-dimensional spectral density [phi]([omega])>=cI>0, [for all][omega][set membership, variant][0,2[pi]] such that [phi]r([omega]) [set membership, variant] Lip* ([alpha]), with covariance block-Toeplitz matrix [Gamma]n of dimension np - np, we show that b=(r+[alpha])/(1+r+[alpha]), [omega]k=2[pi]k/n, (k=l,...,n). This result has applications in extimation of time series and in system identification. We comment how to use this result to derive frequency domain expressions for moltivariate autoregressive spectral density estimates as the order and the number of observations tend to infinity.