{"paper":{"title":"Sharp bounds on the rate of convergence of the empirical covariance matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Alain Pajor, Alexander E. Litvak, Nicole Tomczak-Jaegermann, Rados{\\l}aw Adamczak","submitted_at":"2010-12-01T20:40:01Z","abstract_excerpt":"Let $X_1,..., X_N\\in\\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \\exp(-c\\sqrt{n}\\r)$ one has $\n  \\sup_{x\\in S^{n-1}} \\Big|\\frac{1/N}\\sum_{i=1}^N (|<X_i, x>|^2 - \\E|<X_i, x>|^2\\r)\\Big|\n  \\leq C \\sqrt{\\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $(<X_i,x>)_{i\\leq N, x\\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\\sqrt n)_{i\\leq N}$ exhibit uniformly a sub-exponential decay. As a con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0294","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"}