{"paper":{"title":"The limit distribution in the $q$-CLT for $q \\ge 1$ is unique and can not have a compact support","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Constantino Tsallis, Sabir Umarov","submitted_at":"2016-09-08T13:40:05Z","abstract_excerpt":"In a paper by Umarov, Tsallis and Steinberg (2008), a generalization of the Fourier transform, called the $q$-Fourier transform, was introduced and applied for the proof of a $q$-generalized central limit theorem ($q$-CLT). Subsequently, Hilhorst illustrated (2009 and 2010) that the $q$-Fourier transform for $q>1$ is not invertible in the space of density functions. Indeed, using an invariance principle, he constructed a family of densities with the same $q$-Fourier transform and noted that \"as a consequence, the $q$-central limit theorem falls short of achieving its stated goal\". The distribu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02425","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"}