Consider sample covariance matrices of the form $Q:=\Sigma^{1/2} X X^\top \Sigma^{1/2}$, where $X=(x_{ij})$ is an $n\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite covariance matrix. We study the limiting behavior of the eigenvectors of $Q$ through the so-called eigenvector empirical spectral distribution $F_{\mathbf v}$, which is an alternative form of empirical spectral distribution with weights given by $|\mathbf v^\top \xi_k|^2$, where $\mathbf v$ is a deterministic unit vector and $\xi_k$ are the eigenvectors of $Q$. We prove a functional central limit theorem for the linear spectral statistics of $F_{\mathbf v}$, indexed by functions with H\"older continuous derivatives. We show that the linear spectral statistics converge to some Gaussian processes both on global scales of order 1 and on local scales that are much smaller than 1 but much larger than the typical eigenvalue spacing $N^{-1}$. Moreover, we give explicit expressions for the covariance functions of the Gaussian processes, where the exact dependence on $\Sigma$ and $\mathbf v$ is identified for the first time in the literature.

翻译：考虑形如 $Q:=\Sigma^{1/2} X X^\top \Sigma^{1/2}$ 的样本协方差矩阵，其中 $X=(x_{ij})$ 是一个 $n\times N$ 随机矩阵，其元素为均值为零、方差为 $N^{-1}$ 的独立随机变量，而 $\Sigma$ 是一个确定的正定协方差矩阵。我们通过所谓的特征向量经验谱分布 $F_{\mathbf v}$ 研究 $Q$ 的特征向量的极限行为，这是一种经验谱分布的替代形式，其权重由 $|\mathbf v^\top \xi_k|^2$ 给出，其中 $\mathbf v$ 是一个确定的单位向量，$\xi_k$ 是 $Q$ 的特征向量。我们证明了关于 $F_{\mathbf v}$ 的线性谱统计量的泛函中心极限定理，该统计量由具有 Hölder 连续导数的函数索引。我们证明，无论是在全局尺度（阶为1）上，还是在远小于1但远大于典型特征值间距 $N^{-1}$ 的局部尺度上，线性谱统计量都收敛到某些高斯过程。此外，我们给出了这些高斯过程的协方差函数的显式表达式，其中首次在文献中明确识别了与 $\Sigma$ 和 $\mathbf v$ 的精确依赖关系。