{"paper":{"title":"The limit of the smallest singular value of random matrices with i.i.d. entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov","submitted_at":"2014-10-23T06:58:08Z","abstract_excerpt":"Let $\\{a_{ij}\\}$ $(1\\le i,j<\\infty)$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m)_{m=1}^\\infty$ satisfy $m/N_m\\longrightarrow z$ for some $z\\in(0,1)$. For each $m\\in{\\mathbb N}$ denote by $A_m$ the $N_m\\times m$ random matrix $(a_{ij})$ $(1\\le i\\le N_m,1\\le j\\le m)$ and let $s_{m}(A_m)$ be its smallest singular value. We prove that the sequence $\\bigl({N_m}^{-1/2} s_{m}(A_m)\\bigr)_{m=1}^\\infty$ converges to $1-\\sqrt{z}$ almost surely. Our result does not require boundedness of any moments of $a_{ij}$'s higher than the $2$-nd and res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6263","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"}