{"paper":{"title":"Reduced critical processes for small populations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Minzhi Liu, Vladimir Vatutin","submitted_at":"2018-01-10T02:13:13Z","abstract_excerpt":"Let $\\left\\{ Z(n),n\\geq 1\\right\\} $ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. Assuming that $0<Z(n)\\leq \\varphi (n)$, where either $\\varphi (n)=an$ for some $a>0$ or $\\varphi (n)=o(n)$ as $n\\rightarrow \\infty $, we study the structure of the process $% \\left\\{ Z(m,n),0\\leq m\\leq n\\right\\} ,$ where $Z(m,n)$ is the number of particles in the process at moment $m\\leq n$ having a positive number of descendants at moment $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03217","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"}