{"paper":{"title":"Asymptotic behaviour of heavy-tailed branching processes in random environments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Wenming Hong, Xiaoyue Zhang","submitted_at":"2018-11-18T12:05:26Z","abstract_excerpt":"Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\\mathbb{E}\\log m(\\xi_{0})=\\infty$. We show that (1) there exists no proper $c_{n}$ such that $\\{Z_{n}/c_{n}\\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\\left(\\bar{\\xi},Z_{n}(\\bar{\\xi})\\right)$ converges almost surely to a random variable $Y(\\bar{\\xi})$, where $Y\\in(0,1)~\\eta$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\\left\\{\\frac{U"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.07317","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"}