Let $\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ denote the sample covariance operator of centered i.i.d.~observations $X_1,\dots,X_n$ in a real separable Hilbert space, and let $\Sigma=\mathbb{E}(X_1\otimes X_1)$. The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$, in settings where the eigenvalues of $\Sigma$ decay as $\lambda_j(\Sigma)\asymp j^{-2\beta}$ for some fixed parameter $\beta>1/2$. Our main result shows that the bootstrap can approximate the distribution of $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$ at a rate of order $n^{-\frac{\beta-1/2}{2\beta+4+\epsilon}}$ with respect to the Kolmogorov metric, for any fixed $\epsilon>0$. In particular, this shows that the bootstrap can achieve near $n^{-1/2}$ rates in the regime of large $\beta$ -- which substantially improves on previous near $n^{-1/6}$ rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinate-free techniques. Moreover, our result holds in greater generality, and we propose a model that is compatible with both elliptical and Mar\v{c}enko-Pastur models in high-dimensional Euclidean spaces, which may be of independent interest.

翻译：设 $\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ 为实可分Hilbert空间中中心化独立同分布观测 $X_1,\dots,X_n$ 的样本协方差算子，且 $\Sigma=\mathbb{E}(X_1\otimes X_1)$。本文旨在理解当 $\Sigma$ 的特征值以 $\lambda_j(\Sigma)\asymp j^{-2\beta}$（其中 $\beta>1/2$ 为固定参数）衰减时，自助法在何种程度上能够近似算子范数误差 $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$ 的分布。主要结果表明：对任意固定的 $\epsilon>0$，在Kolmogorov度量下，自助法能以 $n^{-\frac{\beta-1/2}{2\beta+4+\epsilon}}$ 阶速率逼近 $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$ 的分布。特别地，这揭示了当 $\beta$ 较大时，自助法可实现接近 $n^{-1/2}$ 的速率——相较于同区域内先前接近 $n^{-1/6}$ 的速率有显著改进。除获得更快的速率外，我们的分析基于一种基于无坐标技术的根本性新视角。此外，该结果具有更广泛的适用性，我们提出的模型与高维欧氏空间中的椭圆模型及Marčenko-Pastur模型均兼容，这可能具有独立的价值。