On the convergence of the spectrum of finite order approximations of stationary time series

• Conditions for L2L2 convergence of the spectral density of AR and MA approximations.
• Convergence in mean of the spectral density based on covariance estimates.
• Convergence of the spectral density at origin for both cases.

This paper is on the asymptotic behavior of the spectral density of finite autoregressive (AR) and moving average (MA) approximations for a wide sense stationary time series. We consider two aspects: convergence of spectral density of moving average and autoregressive approximations when the covariances are known and when they are estimated. Under certain mild conditions on the spectral density and the covariance sequence, it is shown that the spectral densities of both approximations converge in L2L2 as the order of approximation increases. It is also shown that the spectral density of AR approximations converges at the origin under the same conditions. Under additional regularity assumptions, we show that similar results hold for approximations from empirical covariance estimates.