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Under certain smoothness and regularity conditions, it is shown that, as $u\\to \\infty$, the excursion probability $\\mathbb{P}\\{\\sup_{t\\in T} X(t)\\ge u \\}$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\\mathbb{E}\\{\\chi(A_u(X,T))\\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic f"},"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":"1502.04414","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-16T03:25:01Z","cross_cats_sorted":[],"title_canon_sha256":"b41cbb61301d8d5b11a16e7750f704b56000298927e1ec905dfa1bbe2a89f9e0","abstract_canon_sha256":"d4aff663a4d9bd4e460b550b51ad1e8766df010b48318fb399a469494d789e4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:59.816984Z","signature_b64":"4Gj5UmORmuAcekKWP7QYeJrzXbcEJueGaSanW0cAMf2IaYuhf4WaB2xAmSVTbS/SYwpKvrKwKEoHkm/yWeKHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f27fc51efad8887135dd3a58c5f633bbc6a9723e3c26a82b74f60662a14779ff","last_reissued_at":"2026-05-18T02:26:59.816569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:59.816569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Excursion Probability of Certain Non-centered Smooth Gaussian Random Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dan Cheng","submitted_at":"2015-02-16T03:25:01Z","abstract_excerpt":"Let $X = \\{X(t): t\\in T \\}$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = \\{t\\in T: X(t)\\geq u\\}$ be the excursion set of $X$ exceeding level $u$. Under certain smoothness and regularity conditions, it is shown that, as $u\\to \\infty$, the excursion probability $\\mathbb{P}\\{\\sup_{t\\in T} X(t)\\ge u \\}$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\\mathbb{E}\\{\\chi(A_u(X,T))\\}$, such that the error is super-exponentially small. 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