Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation

The class of nonnegative definite Toeplitz matrices that can be embedded in nonnegative definite circulant matrices of a larger size is characterized. An equivalent characterization in terms of the spectrum of the underlying process is also presented, together with the corresponding extremal processes. It is shown that a given finite-duration sequence ρ can be extended to be the covariance of a periodic stationary processes whenever the Toeplitz matrix R generated by this sequence is strictly positive definite. The sequence ρ=1, cos α, cos 2α with (α/π) irrational, which has a unique nonperiodic extension as a covariance sequence, demonstrates that the strictness is needed. A simple constructive proof supplies a bound on the abovementioned period in terms of the minimal eigenvalue of R. It also yields, under the same conditions, an extension of ρ to covariances that eventually decay to zero. For the maximum-likelihood estimate of the covariance of a stationary Gaussian process, the extension length required for using the estimate-maximize iterative algorithm is determined