{"paper":{"title":"On a class of norms generated by nonnegative integrable distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gilles Stupfler, Michael Falk","submitted_at":"2018-08-24T16:30:54Z","abstract_excerpt":"We show that any distribution function on $\\mathbb{R}^d$ with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on $\\mathbb{R}^{d+1}$, called $F$-norm. We characterize the set of $F$-norms and prove that pointwise convergence of a sequence of $F$-norms to an $F$-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an $F$-norm can easily be estimated by an empirical $F$-norm, whose consistency and weak convergence we establish.\n  The concept of $F$-norms can be extended to arbitrary"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08200","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"}