{"paper":{"title":"A sharpening of Tusn\\'ady's inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"G\\'abor Tusn\\'ady, Jen\\H{o} Reiczigel, L\\'idia Rejt\\H{o}","submitted_at":"2011-10-17T10:34:50Z","abstract_excerpt":"Let ~$\\veps_1, ..., \\veps_m$ be i.i.d. random variables with $$P(\\veps_i=1)= P(\\veps_i= -1)=1/2,$$ and $X_m = \\sum_{i=1}^m \\veps_i.$ Let $Y_m $ be a normal random variable with the same first two moments as that of $X_m.$ There is a uniquely determined function $\\Psi_m$ such that the distribution of $\\Psi_m(Y_m)$ equals to the distribution of $X_m$. Tusn\\'ady's inequality states that $$ \\mid \\Psi_m(Y_m) - Y_m \\mid \\leq \\frac{Y_m^2}{m}+1.$$ Here we propose a sharpened version of this inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3627","kind":"arxiv","version":3},"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"}