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

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 was constructed to fit the first given 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 converges for to a generating function of an original process uniformly on every closed disk , if the original process is of asymptotically constant rank. We shall also show that the sequence of the integrated power spectra of the AR processes converges a.e. for to the integrated power spectrum of an original process even in the degenerate-rank case.