{"paper":{"title":"Is the Sibuya distribution a progeny?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"G\\'erard Letac","submitted_at":"2018-08-27T06:59:52Z","abstract_excerpt":"For $0<a<1$ the Sibuya distribution $s_a$ is concentrated on the set $\\mathbb{N}^+$ of positive integers and is defined by the generating function $\\sum_{n=1}^{\\infty}s_a(n)z^n=1-(1-z)^a.$ A distribution $q$ on $\\mathbb{N}^+$ is called a progeny if there exists a Galton-Watson process\n  $(Z_n)_{n\\geq 0}$ such that $Z_0=1$, such that $\\mathbb{E}(Z_1)\\leq 1$ and such that $q$ is the distribution of $\\sum _{n=0}^{\\infty}Z_n. $ The paper proves that $s_a$ is a progeny if and only if $\\frac{1}{2}\\leq a<1.$ The point is to find the values of $b=1/a$ such that the power series expansion of $u(1-(1-u)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08704","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"}