{"paper":{"title":"Convergence rates for a branching process in a random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chunmao Huang, Quansheng Liu","submitted_at":"2010-10-28T23:37:45Z","abstract_excerpt":"Let $(Z_n)$ be a supercritical branching process in a random environment $\\xi$. We study the convergence rates of the martingale $W_n = Z_n/ E[Z_n| \\xi]$ to its limit $W$. The following results about the convergence almost sur (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \\log W_1^{\\alpha+1} < \\infty$ for some $\\alpha >0$, we have $W-W_n = o (n^{-\\alpha})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second mom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.6111","kind":"arxiv","version":2},"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"}