# Some limit theorems for Szego polynomials

#### Abstract

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. ^

#### Subject Area

Mathematics|Engineering, Electronics and Electrical

#### Recommended Citation

Michael Joseph Arciero,
"Some limit theorems for Szego polynomials"
(2001).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3025528.

http://digitalcommons.uri.edu/dissertations/AAI3025528