{"paper":{"title":"Some properties of the free stable distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.PR","authors_text":"Min Wang, Takahiro Hasebe, Thomas Simon","submitted_at":"2018-05-03T06:08:53Z","abstract_excerpt":"We investigate certain analytical properties of the free $\\alpha-$stable densities on the line. We prove that they are all classically infinitely divisible when $\\alpha\\le 1$, and that they belong to the extended Thorin class when $\\alpha \\leq 3/4.$ The L\\'evy measure is explicitly computed for $\\alpha =1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $\\alpha > 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01133","kind":"arxiv","version":2},"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"}