{"paper":{"title":"Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Daniel Vogel, Dietmar Ferger","submitted_at":"2015-06-13T21:42:03Z","abstract_excerpt":"We prove the asymptotic independence of the empirical process $\\alpha_n = \\sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $\\beta_n = n (F_n(\\tau+\\frac{\\cdot}{n})-F_n(\\tau))$, where $F$ is an arbitrary cdf, differentiable at some point $\\tau$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \\in N$, there is a deterministic correspondence between $\\alpha_n$ and $\\beta_n$. Precisely, we show that the pair $(\\alpha_n,\\beta_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04324","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"}