{"paper":{"title":"Concentration inequalities and asymptotic results for ratio type empirical processes","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Evarist Gin\\'e, Vladimir Koltchinskii","submitted_at":"2006-06-30T09:58:50Z","abstract_excerpt":"Let $\\mathcal{F}$ be a class of measurable functions on a measurable space $(S,\\mathcal{S})$ with values in $[0,1]$ and let \\[P_n=n^{-1}\\sum_{i=1}^n\\delta_{X_i}\\] be the empirical measure based on an i.i.d. sample $(X_1,...,X_n)$ from a probability distribution $P$ on $(S,\\mathcal{S})$. We study the behavior of suprema of the following type: \\[\\sup_{r_n<\\sigma_Pf\\leq \\delta_n}\\frac{|P_nf-Pf|}{\\phi(\\sigma_Pf)},\\] where $\\sigma_Pf\\ge\\operatorname {Var}^{1/2}_Pf$ and $\\phi$ is a continuous, strictly increasing function with $\\phi(0)=0$. Using Talagrand's concentration inequality for empirical pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606788","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"}