{"paper":{"title":"Sharp small-deviation inequalities for sums of independent nonnegative random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Guanyang Wang, Jun Yan, Peng Zhang, Weibo Fu, Yanjun Han, Zhengqing Zhou","submitted_at":"2026-07-27T04:06:33Z","abstract_excerpt":"Let $(X_1,\\ldots,X_n)$ be independent nonnegative random variables with $\\mathbb{E} X_i\\le1$, and write $S=\\sum_iX_i$. For $\\delta>0$, we prove that \\[\n  \\mathbb{P}\\left(S<\\mathbb{E} S+\\delta\\right)\\ge b_{n,\\delta}, \\] where $b_{n,\\delta}=\\delta(n/(n+\\delta))^n$ for $0<\\delta<1$ and $b_{n,\\delta}=(1-1/(n+\\delta))^n$ for $\\delta\\ge1$. The bound is sharp for every $n$ and $\\delta\\ge 1$. In particular, since $b_{n,\\delta} \\ge e^{-1}$ for $\\delta \\ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $\\delta\\ge 1$.\n  The proof is found by ChatGPT 5.6 Pro. It combines the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23980","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/2607.23980/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"}