{"paper":{"title":"Bounding quantiles of Wasserstein distance between true and empirical measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Johannes Wiesel, Martin N. A. Tegn\\'er, Samuel N. Cohen","submitted_at":"2019-07-03T16:01:16Z","abstract_excerpt":"Consider the empirical measure, $\\hat{\\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\\mathbb{P}$ on the unit interval. For fixed $\\mathbb{P}$ the Wasserstein distance between $\\hat{\\mathbb{P}}_N$ and $\\mathbb{P}$ is a random variable on the sample space $[0,1]^N$. Our main result is that its normalised quantiles are asymptotically maximised when $\\mathbb{P}$ is a convex combination between the uniform distribution supported on the two points $\\{0,1\\}$ and the uniform distribution on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.02006","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"}