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We prove that there are positive $u,v,N_0$ depending only on $\\beta$ and $\\delta$ with the following property: for any $N,n$ such that $N\\ge \\max(N_0,\\delta n)$, any $N\\times n$ random matrix $A=(a_{ij})$ with i.i.d. entries satisfying $\\sup\\limits_{\\lambda\\in {\\mathbb R}}{\\mathbb P}\\bigl\\{|a_{11}-\\lambda|\\le 1\\bigr\\}\\le 1-\\beta$ and any non-random $N\\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies ${\\mathbb P}\\bigl\\{s_n(A+B)\\le u\\sqrt{N}\\bigr\\}\\le \\exp(-vN)$. The result holds without any moment assumptions on distrib"},"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":"1409.7975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-29T00:53:01Z","cross_cats_sorted":[],"title_canon_sha256":"ce2db8b368a5abcfb2c9d1352277636cded9fb88827c65c04bb3e4de09f0d5bf","abstract_canon_sha256":"bbd10f8b4f9f8c232cec7f0b376d3ad9dbb067681d3f612ceb4eac476bf3b68a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:33.936428Z","signature_b64":"uq7TKpyC2y1KhgerNZAF2c+AvuNVvIaV1Xr2M85j3qy+lThMxj8C1Rk6kNWBPQ26/YazlnSA/NhuRKgjiGPiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4b6d2d51e9490cf7e7f6fc603304bd5d42bffc2369b6c6eac74d214601773a2","last_reissued_at":"2026-05-18T02:41:33.936011Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:33.936011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The smallest singular value of random rectangular matrices with no moment assumptions on entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konstantin E. Tikhomirov","submitted_at":"2014-09-29T00:53:01Z","abstract_excerpt":"Let $\\delta>1$ and $\\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\\beta$ and $\\delta$ with the following property: for any $N,n$ such that $N\\ge \\max(N_0,\\delta n)$, any $N\\times n$ random matrix $A=(a_{ij})$ with i.i.d. entries satisfying $\\sup\\limits_{\\lambda\\in {\\mathbb R}}{\\mathbb P}\\bigl\\{|a_{11}-\\lambda|\\le 1\\bigr\\}\\le 1-\\beta$ and any non-random $N\\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies ${\\mathbb P}\\bigl\\{s_n(A+B)\\le u\\sqrt{N}\\bigr\\}\\le \\exp(-vN)$. 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