Some limit theorems for Szegö polynomials
We investigate a variety of convergence phenomena for measures on the unit circle associated with certain discrete time stationary stochastic processes, and for the class of Szegö polynomials orthogonal with respect to such measures.
Szegö polynomials, which form the basis of autoregressive (AR) methods in spectral analysis, are not uniquely defined when the degree is less than the number of points on which the spectral measure is supported; that is, when the spectral measure corresponds to a sum of complex sinusoids, the number of which is less than the degree. We consider the asymptotic behavior of Szegö polynomials of fixed degree for certain sequences of measures which converge weakly to such a sum of point masses.
The sequence of measures can be formed in various ways, one of which is by convolving point mass sums with approximate identities, or kernels. In signal processing applications, this corresponds to “windowing” a signal composed of complex sinusoids. The Poisson and Fejër kernels are considered. Another way to form the measures is to add an absolutely continuous measure to a sum of point masses, thus obtaining a spectral measure for sinusoids with additive noise, where the noise coloration is described by the density of the absolutely continuous part. We characterize a limit polynomial for several different classes of sequences of measures. Some special cases are used to interpret research done by others in the field.
Situations where the polynomial degree approaches infinity are considered for fixed measures with a rational spectral density. These measures are the spectral measures for autoregressive moving average (ARMA) random processes. We study the asymptotic behaviors of the reflection coefficients, or constant terms, of the polynomials, and the zero-distribution measures, which consist of point masses at each of the polynomial zeros. These analyses help describe the behavior of the “non-signal” zeros observed in some signal processing situations.
0544: Electrical engineering