On the convergence of the spectral density of autoregressive approximations via empirical covariance estimates

The problem of estimating discrete time stochastic processes by autoregressive models is encountered in many applications. In most practical scenarios, the autoregressive model is derived using estimated values of the covariance sequence (known as the sample covariance) in lieu of the actual covariance sequence of the process. The present paper explores the asymptotic behavior of the spectral density of such approximations, as both the number of samples N and the model order p approach infinity. It is shown that under certain mild assumptions, when p = o{N^1/3}, spectral density of the approximating autoregressive sequence converges at the origin in mean. It is also shown that under the same condition, the spectral density of the autoregressive approximation converges in mean with respect to an L² norm.