Spectral measure of empirical autocovariance matrices of high-dimensional Gaussian stationary processes

Abstract

Consider the empirical autocovariance matrices at given non-zero time lags, 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 measure 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 a shifted version of the autocovariance matrix is obtained. This is then used to obtain the asymptotic behavior 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.