Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class covariance operator $\Sigma$, the problem of recovering the spectral projectors of $\Sigma$ naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator $\hat \Sigma$, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of $\Sigma$. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject only to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.

翻译：给定希尔伯特空间中具有零均值与迹类协方差算子$\Sigma$的有限独立同分布样本，恢复$\Sigma$的谱投影算子问题自然出现在众多应用中。本文研究经验协方差算子$\hat \Sigma$的谱投影算子的分布逼近问题，提出了一个以$\Sigma$的相对秩表征复杂度的维度无关框架。在此设定下，仅需基于矩条件与谱衰减的温和假设，即可建立新型定量极限定理与自助法逼近结果。在诸多情形中，这些结果甚至超越了现有高斯设定下的相关结论。