{"paper":{"title":"How many families survive for a long time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"E.E. Dyakonova, V.A. Vatutin","submitted_at":"2016-08-29T14:18:10Z","abstract_excerpt":"A critical branching process $\\left\\{Z_{k},k=0,1,2,...\\right\\} $ in a random environment generated by a sequence of independent and identically distributed random reproduction laws is considered.\\ Let $Z_{p,n}$ be the number of particles at time $p\\leq n$ having a positive offspring number at time $n$. \\ A theorem is proved describing the limiting behavior, as $% n\\rightarrow \\infty $ of the distribution of a properly scaled process $\\log Z_{p,n}$ under the assumptions $Z_{n}>0$ and $p\\ll n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08062","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"}