{"paper":{"title":"Normal approximation and concentration of spectral projectors of sample covariance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Karim Lounici, Vladimir Koltchinskii","submitted_at":"2015-04-28T02:38:44Z","abstract_excerpt":"Let $X,X_1,\\dots, X_n$ be i.i.d. Gaussian random variables in a separable Hilbert space ${\\mathbb H}$ with zero mean and covariance operator $\\Sigma={\\mathbb E}(X\\otimes X),$ and let $\\hat \\Sigma:=n^{-1}\\sum_{j=1}^n (X_j\\otimes X_j)$ be the sample (empirical) covariance operator based on $(X_1,\\dots, X_n).$ Denote by $P_r$ the spectral projector of $\\Sigma$ corresponding to its $r$-th eigenvalue $\\mu_r$ and by $\\hat P_r$ the empirical counterpart of $P_r.$ The main goal of the paper is to obtain tight bounds on $$ \\sup_{x\\in {\\mathbb R}} \\left|{\\mathbb P}\\left\\{\\frac{\\|\\hat P_r-P_r\\|_2^2-{\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07333","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}