On the asymptotic behavior of the spectral density of autoregressive estimates

The problem of estimating discrete time stochastic processes by autoregressive (AR) models is encountered in many applications. The present paper explores the asymptotic behavior of the spectral density of such approximations. It is shown that under certain assumptions on the spectral density and the covariance sequence of the original process, the spectral density of the approximating autoregressive sequence converges at the origin. Under additional mild conditions it is also shown that the spectral density of the autoregressive approximation converges in L² as the order of approximation increases.