{"paper":{"title":"HCM Property and the Half-Cauchy Distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pierre Bosch (LPP)","submitted_at":"2014-02-05T14:38:23Z","abstract_excerpt":"Let $Z_\\al$ be a positive $\\alpha$-stable random variable and $T_\\al=(Z_\\al/\\tilde Z_\\al)^\\al,$ with independents components in the quotient. It is known that $T_\\al$ is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation $T_\\al^\\beta$ is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if $\\al\\le1/2$ and $|\\beta|\\ge 1/(1-\\al).$ This clarifies a conjecture of Bondesson (1992) on positive stable densities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1059","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"}