Autoregressive model fitting for multichannel time series of degenerate rank: Limit properties

Abstract

The method of fitting an autoregressive (AR) process to a multichannel time series was extended from the full-rank case over the degenerate-rank case in the previous papers [5], [6], and the autoregressive (AR) model \hat{H}_{n}(z) was constructed to fit the first given n + 1 data of an autocorrelation sequence. The notion of random processes of asymptotically constant rank will be introduced in the degenerate-rank case. We shall show that the sequence of the AR models \{ \hat{H}_{n}(z)\} converges for n \rightarrow \infty to a generating function H(z) of an original process uniformly on every closed disk |z| \leq \rho , if the original process is of asymptotically constant rank. We shall also show that the sequence of the integrated power spectra \{ \hat{S}_{n}(e^{j \omega})\} of the AR processes converges a.e. for n \rightarrow \infty to the integrated power spectrum S(e^{j \omega} of an original process even in the degenerate-rank case.