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, which may be of independent interest.

中文翻译：

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提高算子范数的 Bootstrap 逼近率：一种无坐标方法

令 $\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ 表示中心独立同分布观察 $X_1,\dots,X_n$ 在实可分中的样本协方差算子希尔伯特空间，令$\Sigma=\mathbf{E}(X_1\otimes X_1)$。本文的重点是了解 bootstrap 可以在多大程度上逼近算子范数误差 $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$ 的分布，其中对于某个固定参数 $\beta>1/2$，$\Sigma$ 的特征值衰减为 $\lambda_j(\Sigma)\asymp j^{-2\beta}$。我们的主要结果表明，bootstrap 可以以 $n^{-\frac{\ beta-1/2}{2\beta+4+\epsilon}}$ 相对于 Kolmogorov 度量，对于任何固定的 $\epsilon>0$。尤其是，这表明 bootstrap 可以在大 $\beta$ 的情况下实现接近 $n^{-1/2}$ 的利率——这大大提高了之前在相同条件下接近 $n^{-1/6}$ 的利率政权。除了获得更快的速率外，我们的分析还利用了基于无坐标技术的根本不同的视角。此外，我们的结果具有更大的普遍性，我们提出了一个与椭圆模型和 Mar\v{c}enko-Pastur 模型兼容的新模型，这可能具有独立的兴趣。