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

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.

Singular value distribution of dense random matrices with block Markovian dependence

“…Let us note that covariance matrices of time series can also be viewed as an instance of the aforementioned study of random matrices with entries sampled from a stochastic process by consideration of the entries of the data matrix. Sample autocovariance matrices of time series have been considered in [11,16,17,48,49,74].…”

“…. , i k , i 1 ), which occurs on the right-hand side of (17), let G i := (V (i), E(i)) denote the induced undirected graph with vertex set V (i) := {i 1 , . .…”

“…. , i k , i 1 ) be a sequence of integers as occurs in the right-hand side of (17). By property (12) in the definition of an approximately uncorrelated random matrix it holds that P (i) = O k (n −r/2 ) whenever r 1 (i) = r. By the part concerning the strict inequality in Lemma 6.5 we have that for any r ∈ Z >0 there are o k (n (k+r)/2+1 ) terms in the right-hand side of (17) with r 1 (i) = r. This means that only the terms with r 1 (i) = 0 survive the normalization by n −1−k/2 :…”