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We say that $\\{\\Lambda_n\\}_n$ has an asymptotic distribution described by $f$, and we write $\\{\\Lambda_n\\}_n\\sim f$, if \\[ \\lim_{n\\to\\infty}\\frac1{d_n}\\sum_{i=1}^{d_n}F(\\lambda_{i,n})=\\frac1{\\mu_d(\\Omega)}\\int_\\Omega F(f({\\boldsymbol x})){\\rm d}{"},"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":"2309.03662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-09-07T11:59:07Z","cross_cats_sorted":["cs.NA","math.PR"],"title_canon_sha256":"e56364ef25b3f1f42790d25612a5bff1e904454d85e9aedf451682521b7f8419","abstract_canon_sha256":"64f46498a9bb9f0ca647de1e3b42827dbeda824d1e9e046dd9d560f637cf723c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:48:40.244550Z","signature_b64":"JLQS7O+8D2T2vxZaCa1V/i7gjl2neXGQvS30A+fs0PcHEKhYY85saoD1PITNzYzS8iFEwWLmdkKgbWiI8g19Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"913a95734d6deeebd57dd54bfe806efbe12a34b576923ed409ab0dcd901dd5d1","last_reissued_at":"2026-07-05T06:48:40.244072Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:48:40.244072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"From asymptotic distribution and vague convergence to uniform convergence, with numerical applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.PR"],"primary_cat":"math.NA","authors_text":"Carlo Garoni, David Meadon, Giovanni Barbarino, Paris Vassalos, Stefano Serra-Capizzano, Sven-Erik Ekstr\\\"om","submitted_at":"2023-09-07T11:59:07Z","abstract_excerpt":"Let $\\{\\Lambda_n=\\{\\lambda_{1,n},\\ldots,\\lambda_{d_n,n}\\}\\}_n$ be a sequence of finite multisets of real numbers such that $d_n\\to\\infty$ as $n\\to\\infty$, and let $f:\\Omega\\subset\\mathbb R^d\\to\\mathbb R$ be a Lebesgue measurable function defined on a domain $\\Omega$ with $0<\\mu_d(\\Omega)<\\infty$, where $\\mu_d$ is the Lebesgue measure in $\\mathbb R^d$. 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