{"paper":{"title":"Absolute continuity of the martingale limit in branching processes in random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ewa Damek, Konrad Kolesko, Nina Gantert","submitted_at":"2018-06-13T09:07:19Z","abstract_excerpt":"We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $\\xi =(\\xi_n)_{n\\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\\mathbb E (Z_n|\\xi ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $\\xi$ the law of $W$ conditioned on the environment $\\xi $ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \\cite{kaplan:1974}, and of course it covers t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04902","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"}