Embedding Non-Negative Definite Toeplitz Matrices in Non-Negative Definite Circulant Matrices, with Application to Covariance Estimation

We characterize the class of Non-Negative Definite Toeplitz matrices which can be embedded in Non-Negative Definite Circulant matrices of larger size. An equivalent characterization in terms of the spectrum of the underlying process is also presented, together with the corresponding extremal processes. We show that a given finite duration sequence p can be extended to be the covariance of a periodic stationary process whenever the Toeplitz matrix R generated by this sequence is strictly Positive Definite. The sequence p = 1, cos alpha, cos 2 alpha with (alpha/pi) irrational, which has a unique, non-periodic, extension as a covariance sequence, demonstrates that the strictness is needed.