The talk will focus on the approximation of a regular stationary time series by finite AR models. This problem has a long and distinguished history going back to the work of Wold, Parzen, Hannan,and Akaike.

I will present a brief overview about the previous results and then I will present new results on the convergence of the spectral density of the finite dimensional approximations as the order goes to infinity. Specifically we show that a sufficient condition is that the spectral density is strictly positive in [ -\pi , \pi ] and that the coefficients of the Wold decomposition are in l1. I will then address the spectrum estimation problem and show that under the assumption that the stationary sequence

is strongly mixing, the order of the AR approximation should scale as o(n^{1/3}) for convergence of the spectral density estimate in mean square, where n denotes the number of observations.

Joint work with S. Datta Gupta (Waterloo) and P. W. Glynn (Stanford).