{"paper":{"title":"Lower deviation and moderate deviation probabilities for maximum of a branching random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hui He, Xinxin Chen","submitted_at":"2018-07-22T09:28:39Z","abstract_excerpt":"Given a super-critical branching random walk on $\\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \\cite{A13} that as $n\\rightarrow\\infty$, $M_n-x^*n+\\frac{3}{2\\theta^*}\\log n$ converges in law for some suitable constants $x^*$ and $\\theta^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\\mathbb{P}\\left(M_n\\leq x^*n-\\frac{3}{2\\theta^*}\\log n-\\ell_n\\right),$$ for any positive sequence $(\\ell_n)$ such that $\\ell_n=O(n)$ and $\\ell_n\\uparrow\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08263","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"}