Sample correlation matrices are employed ubiquitously in statistics. However, quite surprisingly, little is known about their asymptotic spectral properties for high-dimensional data, particularly beyond the case of "null models" for which the data is assumed independent. Here, considering the popular class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both the leading eigenvalues and eigenvectors of sample correlation matrices. These results are obtained under high-dimensional settings for which the number of samples n and variables p approach infinity, with p/n tending to a constant. To first order, the spectral properties of sample correlation matrices are seen to coincide with those of sample covariance matrices; however their asymptotic distributions can differ significantly, with fluctuations of both the sample eigenvalues and eigenvectors often being remarkably smaller than those of their sample covariance counterparts.

中文翻译：

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高维尖峰模型样本相关矩阵特征结构的渐近性

样本相关矩阵在统计学中无处不在。然而，令人惊讶的是，对于高维数据的渐近谱特性知之甚少，尤其是在假设数据独立的“空模型”之外。在这里，考虑到流行的尖峰模型类，我们应用随机矩阵理论来推导样本相关矩阵的前导特征值和特征向量的渐近一阶和分布结果。这些结果是在高维设置下获得的，其中样本数 n 和变量 p 接近无穷大，p/n 趋于常数。首先，样本相关矩阵的频谱特性与样本协方差矩阵的频谱特性一致；