{"paper":{"title":"Sample covariance matrices of heavy-tailed distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov","submitted_at":"2016-06-11T05:49:36Z","abstract_excerpt":"Let $p>2$, $B\\geq 1$, $N\\geq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $\\sup\\limits_{a\\in S^{n-1}}{\\mathrm E}|\\langle X,a\\rangle|^p\\leq B$. Further, let $X_1,X_2,\\dots,X_N$ be independent copies of $X$, and $\\Sigma_N:=\\frac{1}{N}\\sum_{i=1}^N X_i {X_i}^T$ be the sample covariance matrix. We prove that $$K^{-1}\\|\\Sigma_N-I_n\\|_{2\\to 2}\\leq\\frac{1}{N}\\max\\limits_{i\\leq N}\\|X_i\\|^2 +\\Bigl(\\frac{n}{N}\\Bigr)^{1-2/p}\\log^4\\frac{N}{n}+\\Bigl(\\frac{n}{N}\\Bigr)^{1-2/\\min(p,4)}$$ with probability at least $1-\\frac{1}{n}$, where $K>0$ depends o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03557","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"}