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It is an often quoted conjecture that perhaps $H_{\\alpha} = 1/\\Gamma(1/\\alpha)$ for all $0 < \\alpha \\leq 2$, but it is also frequently observed that this doesn't seem compatible with evidence coming from simulations.\n  We prove the conjecture is false for small $\\alpha$, and in fact that $H_{\\alpha} \\geq (1.1527)^{1/\\alpha}/\\Gamma(1/\\alpha)$ for all sufficiently small $\\alpha$. The proof is a refinement of the \"conditioning and comparison\" approach to lower boun"},"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":"1404.5505","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-22T14:06:18Z","cross_cats_sorted":[],"title_canon_sha256":"2a59ae0e257991e5eb0f258c1ad8d0e07bc39f6dad281e48f7489101122bf793","abstract_canon_sha256":"6834ee4a62c5fbe22f6f41b520302197f557b95678d5ae80d74e81338bcae211"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:31.197474Z","signature_b64":"fWOCRRs+Bh5lqZVIQM60FZEz3fdYt1mAf92sM8lF+g0iNaJXxtuBNJ6FjuDNfPzugYuuVCs6f/ZNT8L0M7kuDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b9c7cb33eb15297f1b1c1ddebd1978749bb0aa2c6c9bb42356240b5b3c8fa62","last_reissued_at":"2026-05-18T02:53:31.196624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:31.196624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pickands' constant $H_{\\alpha}$ does not equal $1/\\Gamma(1/\\alpha)$, for small $\\alpha$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Adam J. Harper","submitted_at":"2014-04-22T14:06:18Z","abstract_excerpt":"Pickands' constants $H_{\\alpha}$ appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps $H_{\\alpha} = 1/\\Gamma(1/\\alpha)$ for all $0 < \\alpha \\leq 2$, but it is also frequently observed that this doesn't seem compatible with evidence coming from simulations.\n  We prove the conjecture is false for small $\\alpha$, and in fact that $H_{\\alpha} \\geq (1.1527)^{1/\\alpha}/\\Gamma(1/\\alpha)$ for all sufficiently small $\\alpha$. 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