Some Limit Theorems for Szego Polynomials

TL;DRAbstract

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 Szego polynomials orthogonal with respect to such measures. Szego 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 Szego 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 Fejer

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