The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal an asymptotic conical structure in critical sample eigendirections under the spike models with distinguishable (or indistinguishable) eigenvalues, when the sample size and/or the number of variables (or dimension) tend to infinity. The consistency of the sample eigenvectors relative to their population counterparts is determined by the ratio between the dimension and the product of the sample size with the spike size. When this ratio converges to a nonzero constant, the sample eigenvector converges to a cone, with a certain angle to its corresponding population eigenvector. In the High Dimension, Low Sample Size case, the angle between the sample eigenvector and its population counterpart converges to a limiting distribution. Several generalizations of the multi-spike covariance models are also explored, and additional theoretical results are presented.

中文翻译：

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高维低样本渐近的统计和数学。

本文的目的是建立多分量尖峰协方差模型主分量分析的几个深层理论特性。我们的新结果揭示了当样本大小和/或变量数量（或维度）趋于无穷大时，具有可区分（或不可区分）特征值的尖峰模型下关键样本特征方向的渐近圆锥结构。样本特征向量相对于其总体对应物的一致性由维度以及样本大小与尖峰大小的乘积之间的比率确定。当该比率收敛到非零常数时，样本特征向量收敛到圆锥体，与其相应的总体特征向量成一定角度。在高维、低样本量的情况下，样本特征向量与其总体对应物之间的角度收敛到极限分布。还探讨了多尖峰协方差模型的几种推广，并提出了额外的理论结果。