{"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 $\\mathbb R^d$. 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}{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.03662","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.03662/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}