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

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Abstract
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

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