{"paper":{"title":"On the anti-concentration functions of some familiar families of distributions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Renming Song, Yuan Tan, Ze-Chun Hu","submitted_at":"2024-01-18T14:14:14Z","abstract_excerpt":"Let $\\{X_{\\alpha}\\}$ be a family of random variables following a certain type of distributions with finite expectation $\\mathbf{E}[X_{\\alpha}]$ and finite variance ${\\rm Var}(X_{\\alpha})$, where $\\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \\infty)\\ni y\\to \\inf_{\\alpha}\\mathbf{P}\\left(|X_{\\alpha}-\\mathbf{E}[X_{\\alpha}]|\\geq y \\sqrt{{\\rm Var}(X_{\\alpha})}\\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, expone"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.09998","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.09998/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"}