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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.6263","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-23T06:58:08Z","cross_cats_sorted":[],"title_canon_sha256":"734b179afd67b24a3537514429c81594f9b72ebd81ecce556b4660eddca22994","abstract_canon_sha256":"b351ec35c8d90ce7906b7ad7e2587518f43e06fe7c171afd5ab9f23a69e75599"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:31.244015Z","signature_b64":"/jMeuK6A/3WsR6LiSPNNfGoG0AM59+uEZmg6AIEMKR8jByQtHR/RFaxlzNLDnjxi76kVvDq0o/s7scPB9Fa2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c772d1b8031a3470a5008af5125792ccc46314cd45977669da0a02a24c91f5a","last_reissued_at":"2026-05-18T02:39:31.243376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:31.243376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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