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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}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2507.09496","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-13T05:12:32Z","cross_cats_sorted":[],"title_canon_sha256":"392cbff5a4930db5ceb99500cbc75f1638bf521a8e30a399cf13ef5704ee5c26","abstract_canon_sha256":"03546d1b397f83cb5a6b578bcc96385678ede81bc3b8b7d16695b61b52994e7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:36:10.085380Z","signature_b64":"9kiTn1zI6ohTPt7YoJ+HQzXVCFDp/PsnZMoDXUjLafoQz1pq0xFZsV50yYqGrJlrKzgmod5WWP2iwLVBh2qXAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4dbe07fa88c738dcb4d072048c570c1a52a9707ac1f434b1ded017856b4ce24d","last_reissued_at":"2026-07-05T11:36:10.084888Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:36:10.084888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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