Spectral measures of empirical autocovariance matrices of high dimensional Gaussian stationary processes

Walid Hachem (1) , Arup Bose

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Keywords: High-dimensional times series analysis; Large non-Hermitian matrix theory; Limit spectral distribution; Matrix orthogonal polynomials; Multivariate stationary processes; Small singular values AMS 2020 Subject Classification: Primary 60B20; Secondary 60G57; 62M10; 33C47

Domains: Probability [math.PR]; Signal and Image Processing

Walid Hachem, Arup Bose. Spectral measures of empirical autocovariance matrices of high dimensional Gaussian stationary processes. 2021. ⟨hal-03446803v1⟩