# Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach

@inproceedings{Lopes2022ImprovedRO, title={Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach}, author={Miles E. Lopes}, year={2022} }

Let Σ̂ = 1 n∑ni=1Xi⊗Xi denote the sample covariance operator of centered i.i.d. observations X1, . . . ,Xn in a real separable Hilbert space, and let Σ =E(X1⊗X1). The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error √ n∥Σ̂ − Σ∥op, in settings where the eigenvalues of Σ decay as λj(Σ) ≍ j−2β for some fixed parameter β > 1/2. Our main result shows that the bootstrap can approximate the distribution of √n∥Σ̂ −Σ∥op at a rate of…

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