{"paper":{"title":"Weighted Gaussian Approximations for Increments of the Uniform Empirical and Quantile Processes: Fixed-Endpoint Extensions to the Finite-Count Scale","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Abdelhakim Necir","submitted_at":"2026-06-26T16:11:52Z","abstract_excerpt":"We establish weighted Gaussian approximations for the uniform empirical and quantile processes and for their increments ending at a fixed point \\(t\\in(0,1)\\). We first place the classical weighted approximations for the ordinary processes in a common framework and then show that the corresponding increment approximations remain valid uniformly down to the finite-count scale \\(\\lambda/n\\), for every fixed \\(\\lambda>0\\). For the empirical increments, the proof splits the sample at \\(t\\), couples the two resulting conditional empirical processes with independent Brownian bridges, and approximates"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28222","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28222/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"}