{"paper":{"title":"Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Elizaveta Rebrova, Konstantin Tikhomirov","submitted_at":"2015-08-26T23:56:06Z","abstract_excerpt":"Let $A=(a_{ij})$ be an $n\\times n$ random matrix with i.i.d. entries such that $\\mathbb{E} a_{11} = 0$ and $\\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $\\delta>0$ there is $L>0$ depending only on $\\delta$, and a subset $\\mathcal{N}$ of $B_2^n$ of cardinality at most $\\exp(\\delta n)$ such that with probability very close to one we have $$A(B_2^n)\\subset \\bigcup_{y\\in A(\\mathcal{N})}\\bigl(y+L\\sqrt{n}B_2^n\\bigr).$$ As an application, we show that for some $L'>0$ and $u\\in[0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06690","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"}