论文标题
大型样品相关矩阵：比较定理及其应用
Large sample correlation matrices: a comparison theorem and its applications
论文作者
论文摘要
在本文中，我们表明，由$ n $ n $独立观察的高维样品协方差矩阵的对角线可以通过人口协方差矩阵的对角线来在光谱规范中近似具有有限的第四次时刻的$ p $维时间序列。我们假设$ n，p \ to \ infty $，$ p/n $趋向于正常或零。作为应用程序，我们提供了样本相关矩阵$ {\ mathbf r} $的近似值，并为其特征值得出了各种结果。我们确定$ {\ mathbf r} $的限制光谱分布，并为人口相关矩阵及其特征值构建估计器。最后，分析了$ {\ mathbf r} $的极端特征值的几乎确定的限制。
In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from $n$ independent observations of a $p$-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that $n,p\to \infty$ with $p/n$ tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix ${\mathbf R}$ and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of ${\mathbf R}$ and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of ${\mathbf R}$ in a generalized spiked correlation model are analyzed.
