On the asymptotic eigenvalue distribution of Toeplitz matrices

Abstract

Since covariance matrices of weakly stationary random processes are Toeplitz, much of the theory involving asymptotic results for such processes is simply the theory of the asymptotic behavior of Toeplitz forms. The fundamental theorem of this type is the Szegö theorem on the asymptotic eigenvalue distribution of Toeplitz matrices. This theorem is often quoted but relatively little understood in the engineering literature. In this tutorial paper we prove the Szegiö theorem for the special case of finite-order Toeplitz matrices. In this setting the mathematical sophistication of the classical proofs is not required and the proof is both simple and intuitive--yet it contains the important concepts involved in the most general case.