{"paper":{"title":"Asymptotics and Concentration Bounds for Bilinear Forms 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":"2014-08-20T13:20:50Z","abstract_excerpt":"Let $X,X_1,\\dots, X_n$ be i.i.d. Gaussian random variables with zero mean and covariance operator $\\Sigma={\\mathbb E}(X\\otimes X)$ taking values in a separable Hilbert space ${\\mathbb H}.$ Let $$ {\\bf r}(\\Sigma):=\\frac{{\\rm tr}(\\Sigma)}{\\|\\Sigma\\|_{\\infty}} $$ be the effective rank of $\\Sigma,$ ${\\rm tr}(\\Sigma)$ being the trace of $\\Sigma$ and $\\|\\Sigma\\|_{\\infty}$ being its operator norm. Let $$\\hat \\Sigma_n:=n^{-1}\\sum_{j=1}^n (X_j\\otimes X_j)$$ be the sample (empirical) covariance operator based on $(X_1,\\dots, X_n).$ The paper deals with a problem of estimation of spectral projectors of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4643","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":""},"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"}