{"paper":{"title":"On large deviation rates for sums associated with Galton-Watson processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hui He","submitted_at":"2015-02-05T04:50:50Z","abstract_excerpt":"Given a super-critical Galton-Watson process $\\{Z_n\\}$ and a positive sequence $\\{\\epsilon_n\\}$, we study the limiting behaviors of $P(S_{Z_n}/Z_n\\geq\\epsilon_n)$ and $P(S_{Z_n}/m^n\\geq\\epsilon_n) $ with sums $S_{n}$ of i.i.d. random variables $X_i$ and $m=E[Z_1]$. We assume that we are in Schr\\\"oder case with $EZ_1\\log Z_1<\\infty$ and $X_1$ is in the domain of attraction of an $\\alpha$-stable law with $0<\\alpha<2$. As by-products, when $Z_1$ is sub-exponentially distributed, we further obtain the convergence rates of $ \\frac{Z_{n+1}}{Z_n}$ to $m$ as $n\\rightarrow\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01433","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"}