Joint CLT for top eigenvalues of sample covariance matrices of separable high dimensional long memory processes

Abstract

For [Formula: see text], consider the sample covariance matrix [Formula: see text] from a data set [Formula: see text], where [Formula: see text] is a [Formula: see text] matrix having i.i.d. entries with mean zero and variance one, and [Formula: see text] are deterministic positive semi-definite Hermitian matrices of dimension [Formula: see text] and [Formula: see text], respectively. We assume that [Formula: see text] is bounded in spectral norm, and [Formula: see text] is a Toeplitz matrix with its largest eigenvalues diverging to infinity. The matrix [Formula: see text] can be viewed as a data set of an [Formula: see text]-dimensional long memory stationary process having separable dependence structure. As [Formula: see text] and [Formula: see text], we establish the asymptotics and the joint CLT for [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th largest eigenvalue of [Formula: see text] and [Formula: see text] is a fixed integer. For the CLT, we first study the case where the entries of [Formula: see text] are Gaussian, and then we generalize the result to some more generic cases. This result substantially extends our previous result in Merlevède et al. 2019, where we studied [Formula: see text] in the case where [Formula: see text] and [Formula: see text] with [Formula: see text] having Gaussian entries. In order to establish this CLT, we are led to study the first order asymptotics of the largest eigenvalues and the associated eigenvectors of some deterministic Toeplitz matrices related to long memory stationary processes. We prove multiple spectral gap properties for the largest eigenvalues and a delocalization property for their associated eigenvectors.