{"paper":{"title":"How big is the minimum of a branching random walk?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yueyun Hu (LAGA)","submitted_at":"2013-05-28T11:22:40Z","abstract_excerpt":"Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \\to \\infty$, $M_n- {3 \\over 2} \\log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the upper limits of $M_n$: upon the system's non-extinction, $ \\limsup\\_{n\\to \\infty} {1\\over \\log \\log \\log n} ( M_n - {3\\over2} \\log n) = 1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n- {3 \\over 2} \\log n > \\lambda)$ for $\\lambda\\to \\infty$ and $\\lambda=o(\\log n)$. This problem is closely related to the small deviations of a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6448","kind":"arxiv","version":5},"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"}