Although there is an extensive literature on the eigenvalues of high-dimensional sample covariance matrices, much of it is specialized to independent components (IC) models -- in which observations are represented as linear transformations of random vectors with independent entries. By contrast, less is known in the context of elliptical models, which violate the independence structure of IC models and exhibit quite different statistical phenomena. In particular, very little is known about the scope of bootstrap methods for doing inference with spectral statistics in high-dimensional elliptical models. To fill this gap, we show how a bootstrap approach developed previously for IC models can be extended to handle the different properties of elliptical models. Within this setting, our main theoretical result guarantees that the proposed method consistently approximates the distributions of linear spectral statistics, which play a fundamental role in multivariate analysis. We also provide empirical results showing that the proposed method performs well for a variety of nonlinear spectral statistics.

翻译：尽管关于高维样本协方差矩阵特征值的研究已有大量文献，但其中许多工作专门针对独立成分（IC）模型——在该模型中，观测值被表示为具有独立分量的随机向量的线性变换。相比之下，对于破坏IC模型独立性结构并展现出截然不同统计现象的椭圆模型，目前了解较少。特别地，关于在高维椭圆模型中基于Bootstrap方法进行谱统计量推断的范围，现有研究极为有限。为填补这一空白，我们展示了如何将先前为IC模型开发的Bootstrap方法扩展至处理椭圆模型的异质性特征。在该框架下，我们的主要理论结果保证了所提方法能一致逼近线性谱统计量的分布——这类统计量在多变量分析中起基础性作用。我们同时提供了实证结果，表明所提方法对多种非线性谱统计量亦表现良好。