{"paper":{"title":"Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Miles E. Lopes","submitted_at":"2022-08-05T09:07:33Z","abstract_excerpt":"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 $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.03050","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.03050/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}