{"paper":{"title":"A Note on Distribution Free Symmetrization Inequalities","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jiange Li, Wenbo V. Li, Zhao Dong","submitted_at":"2014-01-07T00:18:00Z","abstract_excerpt":"Let $X, Y$ be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space $(\\mathcal{X}, \\|\\cdot\\|)$. Given two measurable subsets $F, K\\subseteq\\cal{X}$, we established distribution free comparison inequalities between $\\mathbb{P}(X\\pm Y \\in F)$ and $\\mathbb{P}(X-Y\\in K)$. These estimates are optimal for real random variables as well as when $\\mathcal{X}=\\mathbb{R}^d$ is equipped with the $\\|\\cdot\\|_\\infty$ norm. Our approach for both problems extends techniques developed by Schultze and Weizs\\\"acher (2007)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1243","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":""},"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"}