For N,n ∈ ℕ, consider the sample covariance matrix SN(T) = 1 NXX∗ from a data set X = CN1/2ZT n1/2, where Z = (Zi,j) is a N × n matrix having i.i.d. entries with mean zero and variance one, and CN,Tn are deterministic positive semi-definite Hermitian matrices of dimension N and n, respectively. We assume that (CN)N is bounded in spectral norm, and Tn is a Toeplitz matrix with its largest eigenvalues diverging to infinity. The matrix X can be viewed as a data set of an N-dimensional long memory stationary process having separable dependence structure. As N,n →∞ and Nn−1 → r ∈ (0,∞), we establish the asymptotics and the joint CLT for (λ1(SN(T)),…,λm(SN(T))⊤, where λj(SN(T)) denotes the jth largest eigenvalue of SN(T) and m is a fixed integer. For the CLT, we first study the case where the entries of Z 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 λ1(SN(T)) in the case where m = 1 and X = ZTn1/2 with Z 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.

中文翻译：

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可分离高维长记忆过程样本协方差矩阵顶部特征值的联合 CLT

为了ñ,n ∈ ℕ，考虑样本协方差矩阵小号ñ(吨) = 1 ñXX*从一个数据集X = Cñ1/2Z吨 n1/2， 在哪里Z = (Z一世,j)是一个ñ × n具有均值为零和方差为 1 的 iid 条目的矩阵，以及Cñ,吨n是确定性的半正定 Hermitian 维数矩阵ñ和n， 分别。我们假设(Cñ)ñ在频谱范数中是有界的，并且吨n是一个 Toeplitz 矩阵，其最大特征值发散到无穷大。矩阵X可以看作是一个数据集ñ具有可分离依赖结构的维长记忆平稳过程。作为ñ,n →∞和ñn-1 → r ∈ (0,∞), 我们建立渐近线和联合 CLT(λ1(小号ñ(吨)),…,λ米(小号ñ(吨))⊤， 在哪里λj(小号ñ(吨))表示j的最大特征值小号ñ(吨)和米是一个固定整数。对于 CLT，我们首先研究Z是高斯的，然后我们将结果推广到一些更一般的情况。这个结果大大扩展了我们之前在 Merlevède 的结果等。2019年，我们学习的地方λ1(小号ñ(吨))在这种情况下米 = 1和X = Z吨n1/2和Z具有高斯条目。为了建立这个 CLT，我们被引导研究与长记忆平稳过程相关的一些确定性 Toeplitz 矩阵的最大特征值和相关特征向量的一阶渐近。我们证明了最大特征值的多个光谱间隙属性和相关特征向量的离域属性。