{"paper":{"title":"Harmonic moments and large deviations for a supercritical branching process in a random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Eric Miqueu, Ion Grama, Quansheng Liu","submitted_at":"2016-08-29T14:38:34Z","abstract_excerpt":"Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\\xi$. We study the asymptotic of the harmonic moments $\\mathbb{E}\\left[Z_n^{-r} | Z_0=k \\right]$ of order $r>0$ as $n \\to \\infty$. We exhibit a phase transition with the critical value $r_k>0$ determined by the equation $\\mathbb E p_1^k = \\mathbb E m_0^{-r_k},$ where $m_0=\\sum_{k=0}^\\infty k p_k$ with $p_k=\\mathbb P(Z_1=k | \\xi),$ assuming that $p_0=0.$ Contrary to the constant environment case (the Galton-Watson case), this critical value is different from that for the existence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08075","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"}