New results in the existence of complex covariance estimates

The problem of generating a positive definite maximum likelihood (ML) estimate of a complex Toeplitz covariance matrix given one N -length data vector is considered. The data vector is assumed to be drawn from a complex Gaussian population with mean zero and covariance σ²I. An upper bound is derived on the measure of the set of such N-length data vectors such that, for one data vector, the ML estimation procedure yields a positive definite solution. These data vectors are denoted as those that do not satisfy the failure condition of the ML estimation procedure, and it is shown that the measure of the set of such data vectors is small and converges to zero as the length of the data vector increases