{"paper":{"title":"Comparing Fr\\'echet and positive stable laws","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"LPTMS), Thomas Simon (LPP","submitted_at":"2013-10-07T19:01:53Z","abstract_excerpt":"Let ${\\bf L}$ be the unit exponential random variable and ${\\bf Z}_\\alpha$ the standard positive $\\alpha$-stable random variable. We prove that $\\{(1-\\alpha) \\alpha^{\\gamma_\\alpha} {\\bf Z}_\\alpha^{-\\gamma_\\alpha}, 0< \\alpha <1\\}$ is decreasing for the optimal stochastic order and that $\\{(1-\\alpha){\\bf Z}_\\alpha^{-\\gamma_\\alpha}, 0< \\alpha < 1\\}$ is increasing for the convex order, with $\\gamma_\\alpha = \\alpha/(1-\\alpha).$ We also show that $\\{\\Gamma(1+\\alpha) {\\bf Z}_\\alpha^{-\\alpha}, 1/2\\le \\alpha \\le 1\\}$ is decreasing for the convex order, that ${\\bf Z}_\\alpha^{-\\alpha}\\,\\prec_{st}\\, \\Gamm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1888","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"}