  Consider the empirical autocovariance matrix at a given non-zero time lag
based on observations from a multivariate complex Gaussian stationary time
series. The spectral analysis of these autocovariance matrices can be useful in
certain statistical problems, such as those related to testing for white noise.
We study the behavior of their spectral measures in the asymptotic regime where
the time series dimension and the observation window length both grow to
infinity, and at the same rate. Following a general framework in the field of
the spectral analysis of large random non-Hermitian matrices, at first the
probabilistic behavior of the small singular values of the shifted versions of
the autocovariance matrix are obtained. This is then used to infer about the
large sample behaviour of the empirical spectral measure of the autocovariance
matrices at any lag. Matrix orthogonal polynomials on the unit circle play a
crucial role in our study.
