{"paper":{"title":"On the self-decomposability of the Fr\\'echet distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Pierre Bosch (LPP), Thomas Simon (LPP)","submitted_at":"2013-02-13T14:05:39Z","abstract_excerpt":"Let $\\{\\Gamma_t, \\, t\\ge 0\\}$ be the Gamma subordinator. Using a moment identification due to Bertoin-Yor (2002), we observe that for every $t > 0$ and $\\alpha\\in (0,1)$ the random variable $\\Gamma_t^{-\\alpha}$ is distributed as the exponential functional of some spectrally negative L\\'evy process. This entails that all size-biased samplings of Fr\\'echet distributions are self-decomposable and that the extreme value distribution $F_\\xi$ is infinitely divisible if and only if $\\xi\\not\\in (0,1),$ solving problems raised by Steutel (1973) and Bondesson (1992). We also review different analytical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.3097","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"}