{"paper":{"title":"An optimal Berry-Esseen type theorem for integrals of smooth functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Irina Shevtsova, Lutz Mattner","submitted_at":"2017-10-23T20:40:11Z","abstract_excerpt":"We prove a Berry-Esseen type inequality for approximating expectations of sufficiently smooth functions $f$, like $f=|\\cdot|^3$, with respect to standardized convolutions of laws $P_1,\\ldots, P_n$ on the real line by corresponding expectations based on symmetric two-point laws $Q_1,\\ldots,Q_n$ isoscedastic to the $P_i$. Equality is attained for every possible constellation of the Lipschitz constant $\\|f\"\\|^{}_{\\mathrm{L}}$ and the variances and the third centred absolute moments of the $P_i$. The error bound is strictly smaller than $\\frac 16$ times the Lyapunov ratio times $\\|f\"\\|^{}_{\\mathrm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08503","kind":"arxiv","version":2},"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"}