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

TL;DR

This paper analyzes the spectral behavior of empirical autocovariance matrices from high-dimensional Gaussian stationary processes, revealing their asymptotic spectral measures as both dimension and sample size grow large.

Contribution

It introduces a framework for understanding the spectral measures of autocovariance matrices in high dimensions, using random matrix theory and matrix orthogonal polynomials.

Findings

Spectral measures converge in the high-dimensional limit.

Small singular values behavior informs spectral distribution.

Framework applicable to statistical testing of white noise.

Abstract

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.