{"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 $\\alpha$. The proof is a refinement of the \"conditioning and comparison\" approach to lower boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5505","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"}