{"paper":{"title":"On a Tail Bound for Root-Finding in Randomly Growing Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"N. D. Shyamalkumar, Sam Justice","submitted_at":"2019-05-18T22:10:14Z","abstract_excerpt":"We re-examine a lower-tail upper bound for the random variable $$X=\\prod_{i=1}^{\\infty}\\min\\left\\{\\sum_{k=1}^iE_k,1\\right\\},$$ where $E_1,E_2,\\ldots\\stackrel{iid}\\sim\\text{Exp}(1)$. This bound has found use in root-finding and seed-finding algorithms for randomly growing trees, and was initially proved as a lemma in the context of the uniform attachment tree model. We first show that $X$ has a useful representation as a compound product of uniform random variables that allows us to determine its moments and refine the existing nonasymptotic bound. Next we demonstrate that the lower-tail probab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.07652","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"}