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.

基于多元复高斯平稳时间序列的观察，考虑给定非零时间滞后的经验自协方差矩阵。这些自协方差矩阵的频谱分析可用于某些统计问题，例如与测试白噪声相关的问题。我们研究了它们在渐近状态下的光谱测量行为，其中时间序列维度和观察窗口长度都以相同的速率增长到无穷大。遵循大型随机非 Hermitian 矩阵的谱分析领域的一般框架，首先获得自协方差矩阵的移位版本的小奇异值的概率行为。然后将其用于获得自协方差矩阵在任何滞后的经验谱测量的渐近行为。