A proof of the pointwise convergence of the multidimensional minimum variance spectral estimator as the region of data support becomes infinite is given. It is shown that an octant is sufficient to ensure that the minimum variance spectral estimator will converge to the true power spectral density. The proof is valid for 1-D, multidimensional, continuous, and mixed spectra. Another useful result is that a normalized minimum variance spectral estimator can be defined to indicate sinusoidal power for processes with a mixed spectrum. Finally, upper and lower bounds on the continuous portion of the spectral estimate are given.

It is shown that the zeros of polynomials orthogonal with respect to certain positive measures on the unit circle have an asymptotic distribution which is uniform on a circle of radius less than or equal to one. These results explain some phenomena observed when various linear prediction methods are used to estimate sinusoidal frequencies. They also describe the asymptotic zero distribution of the prediction error filter polynomials for a class of time series including autoregressive moving average (ARMA) models.

A stationary Gaussian process\xi_{t}, with a spectral densityfsatisfying\int (\log f)(1 + x_{2})^{-1} dx > \inftyagrees in a finite interval [- T,T] with a process et which has a purely discrete spectral measure.