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 $\Sigma=\mathbf{E}(X_1\otimes X_1)$. 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 new 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：令$\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$表示实可分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$\Sigma=\mathbf{E}(X_1\otimes X_1)$。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，自举法对算子范数误差$\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$分布的逼近效果。我们的主要结果表明，对任意固定$\epsilon>0$，自举法能以Kolmogorov度量意义下$n^{-\frac{\beta-1/2}{2\beta+4+\epsilon}}$的速率逼近$\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$的分布。特别地，这显示在$\beta$较大的情况下，自举法可实现接近$n^{-1/2}$的速率——相较于先前同一情景下接近$n^{-1/6}$的速率有显著提升。除获得更快速率外，我们的分析还基于一种基于无坐标技术的根本性不同视角。此外，我们的结果在更广泛的意义下成立，并提出了一种与高维欧氏空间中椭圆模型和Marčenko-Pastur模型均兼容的新模型，该模型可能具有独立的研究价值。