{"paper":{"title":"Berry-Esseen's bound and Cram\\'er's large deviation expansion 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":"2016-02-05T16:12:32Z","abstract_excerpt":"Let $(Z_n)$ be a supercritical branching process in a random environment $\\xi = (\\xi_n)$. We establish a Berry-Esseen bound and a Cram\\'er's type large deviation expansion for $\\log Z_n$ under the annealed law $\\mathbb P$. We also improve some earlier results about the harmonic moments of the limit variable $W=lim_{n\\to \\infty} W_n$, where $W_n =Z_n/ \\mathbb{E}_{\\xi} Z_n$ is the normalized population size."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02081","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"}