论文标题
各向异性样品协方差矩阵的特征向量的线性光谱统计
Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices
论文作者
论文摘要
考虑表格$ q的样本协方差矩阵：=σ^{1/2} x x x^\ topσ^{1/2} $，其中$ x =（x_ {ij}）$是$ n \ times n $ n $随机矩阵的条目，其条目是带有平均零和方差$ n^is a的独立变量usianter $ n^^$ a a和$ n^= - 和$ 1} $ a a，以及$ n^{-1} $ = -1} $ = -1} $ = -1} $σ 矩阵。我们通过所谓的特征向量的经验光谱分布$ f _ {\ Mathbf v} $研究$ q $的特征向量的限制行为，这是经验光谱分布的另一种形式，具有$ | \ m m i | \ mathbf V^\ top top top top top top top top top top top top top topectian $ e^2 $ nution $ etmatian $ \ eastion $ $ q $的特征向量。我们证明了$ f _ {\ Mathbf V} $的线性光谱统计量的功能性中心限制定理，该函数由Hölder连续导数的函数索引。我们表明，线性频谱统计量在订单1的全局尺度和局部量表上都融合到某些高斯过程，这些量表远小于1，但比典型的特征值间距$ n^{ - 1} $大得多。此外，我们为高斯流程的协方差函数提供了明确的表达式，其中的确切依赖性在文献中是首次确定的。
Consider sample covariance matrices of the form $Q:=Σ^{1/2} X X^\top Σ^{1/2}$, 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 $Σ$ 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 $|\mathbf v^\top ξ_k|^2$, where $\mathbf v$ is a deterministic unit vector and $ξ_k$ 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 Hölder 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 $Σ$ and $\mathbf v$ is identified for the first time in the literature.
