{"paper":{"title":"Revisit on the convergence rate of normal extremes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bingjie Tian, Yutao ma","submitted_at":"2025-07-13T05:12:32Z","abstract_excerpt":"Let $(X_i)_{1 \\le i \\le n}$ be independent and identically distributed (i.i.d.) standard Gaussian random variables, and denote by $X_{(n)} = \\max_{1 \\le i \\le n} X_i$ the maximum order statistic. It is well-known in extreme value theory that the linearly normalized maximum $\nY_n = a_n(X_{(n)} - b_n), $\nconverges weakly to the standard Gumbel distribution $\\Lambda$ as $n \\to \\infty$, where $a_n > 0$ and $b_n$ are appropriate scaling and centering constants. In this note, choosing $$a_n=\\sqrt{2\\log n}\\quad \\text{and}\\quad b_n = \\sqrt{2 \\log n} - \\frac{\\log \\log n + \\log (4\\pi)}{2 \\sqrt{2 \\log n}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.09496","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.09496/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}