{"paper":{"title":"Remarks on Talagrand's deviation inequality for Rademacher functions","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Gideon Schechtman, William B. Johnson","submitted_at":"1990-02-16T00:00:00Z","abstract_excerpt":"Recently Talagrand [T] estimated the deviation of a function on $\\{0,1\\}^n$ from its median in terms of the Lipschitz constant of a convex extension of $f$ to $\\ell ^n_2$; namely, he proved that\n  $$P(|f-M_f| > c) \\le 4 e^{-t^2/4\\sigma ^2}$$ where $\\sigma$ is  the Lipschitz constant of the extension of $f$ and $P$ is the natural probability on $\\{0,1\\}^n$.\n  Here we extend this inequality to more general product probability spaces; in particular, we prove the same inequality for $\\{0,1\\}^n$ with the product measure $((1-\\eta)\\delta _0 + \\eta \\delta _1)^n$. We believe this should be useful in p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9201208","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"}