{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:2OPVUHYC3HRTN3X5B64TYEUKY2","short_pith_number":"pith:2OPVUHYC","canonical_record":{"source":{"id":"1211.6693","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-28T18:21:43Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"93a8920d37bc209661d6b295cafa83e1d1d73deb2e4b5e83d426c7bcc9439732","abstract_canon_sha256":"a088c45d0f63ffea4b4d2e74ce4ae07a0df4afe4e1b12247c0ca5bc48e95851d"},"schema_version":"1.0"},"canonical_sha256":"d39f5a1f02d9e336eefd0fb93c128ac6a8f1ea91c9727f1ad37311008a48f2b3","source":{"kind":"arxiv","id":"1211.6693","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6693","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6693v3","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6693","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"pith_short_12","alias_value":"2OPVUHYC3HRT","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2OPVUHYC3HRTN3X5","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2OPVUHYC","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:2OPVUHYC3HRTN3X5B64TYEUKY2","target":"record","payload":{"canonical_record":{"source":{"id":"1211.6693","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-28T18:21:43Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"93a8920d37bc209661d6b295cafa83e1d1d73deb2e4b5e83d426c7bcc9439732","abstract_canon_sha256":"a088c45d0f63ffea4b4d2e74ce4ae07a0df4afe4e1b12247c0ca5bc48e95851d"},"schema_version":"1.0"},"canonical_sha256":"d39f5a1f02d9e336eefd0fb93c128ac6a8f1ea91c9727f1ad37311008a48f2b3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:39.238516Z","signature_b64":"crOG+5uzkHk0jrXl2COtM3SweNHR//XYh+uEy95QklL6AM1s+/yjhOVIO4lg3QJ7vnkeeAaNzyXhEQldqleTDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d39f5a1f02d9e336eefd0fb93c128ac6a8f1ea91c9727f1ad37311008a48f2b3","last_reissued_at":"2026-05-18T01:15:39.237728Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:39.237728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.6693","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4A2MbvD9OoXNFRg109i8PLbOIdoZPh1h/UT89h9n7MVArNcaxvUGcOWOz8nDg9LgDuNFdJlVCCrHp8PQFRhxAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:04:28.246781Z"},"content_sha256":"312d8824743764bfa3ba24cc094db200e99a73a0a09ef2513ba478265619c124","schema_version":"1.0","event_id":"sha256:312d8824743764bfa3ba24cc094db200e99a73a0a09ef2513ba478265619c124"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:2OPVUHYC3HRTN3X5B64TYEUKY2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Dan Cheng, Yimin Xiao","submitted_at":"2012-11-28T18:21:43Z","abstract_excerpt":"Let $X=\\{X(t),t\\in {\\mathbb{R}}^N\\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\\subset {\\mathbb{R}}^N$ and $u\\in {\\mathbb{R}}$, denote by $A_u=\\{t\\in T:X(t)\\geq u\\}$ the excursion set. Under $X(\\cdot)\\in C^2({\\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\\mathbb{E}}\\{\\varphi(A_u)\\}$, is derived. By applying the Rice method, it is shown that, as $u\\to\\infty$, the excursion probability ${\\mathbb{P}}\\{\\sup_{t\\in T}X(t)\\geq u\\}$ can be approximated by ${\\mathbb{E}}\\{\\varphi(A_u)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6693","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EbJ10FWKmuvuScanmg40AfMR63FyvUucGe0v7ML/JWmpUg4v16nMgCqEVCwt05LAQnF92Zy370H9eGxWVVmqDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:04:28.247411Z"},"content_sha256":"fcb614248560c33db8b1b4b39133651b4979fdfc7a1864fa338dd4bab64d6333","schema_version":"1.0","event_id":"sha256:fcb614248560c33db8b1b4b39133651b4979fdfc7a1864fa338dd4bab64d6333"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2OPVUHYC3HRTN3X5B64TYEUKY2/bundle.json","state_url":"https://pith.science/pith/2OPVUHYC3HRTN3X5B64TYEUKY2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2OPVUHYC3HRTN3X5B64TYEUKY2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T05:04:28Z","links":{"resolver":"https://pith.science/pith/2OPVUHYC3HRTN3X5B64TYEUKY2","bundle":"https://pith.science/pith/2OPVUHYC3HRTN3X5B64TYEUKY2/bundle.json","state":"https://pith.science/pith/2OPVUHYC3HRTN3X5B64TYEUKY2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2OPVUHYC3HRTN3X5B64TYEUKY2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:2OPVUHYC3HRTN3X5B64TYEUKY2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a088c45d0f63ffea4b4d2e74ce4ae07a0df4afe4e1b12247c0ca5bc48e95851d","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-28T18:21:43Z","title_canon_sha256":"93a8920d37bc209661d6b295cafa83e1d1d73deb2e4b5e83d426c7bcc9439732"},"schema_version":"1.0","source":{"id":"1211.6693","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6693","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6693v3","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6693","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"pith_short_12","alias_value":"2OPVUHYC3HRT","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2OPVUHYC3HRTN3X5","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2OPVUHYC","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:fcb614248560c33db8b1b4b39133651b4979fdfc7a1864fa338dd4bab64d6333","target":"graph","created_at":"2026-05-18T01:15:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $X=\\{X(t),t\\in {\\mathbb{R}}^N\\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\\subset {\\mathbb{R}}^N$ and $u\\in {\\mathbb{R}}$, denote by $A_u=\\{t\\in T:X(t)\\geq u\\}$ the excursion set. Under $X(\\cdot)\\in C^2({\\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\\mathbb{E}}\\{\\varphi(A_u)\\}$, is derived. By applying the Rice method, it is shown that, as $u\\to\\infty$, the excursion probability ${\\mathbb{P}}\\{\\sup_{t\\in T}X(t)\\geq u\\}$ can be approximated by ${\\mathbb{E}}\\{\\varphi(A_u)","authors_text":"Dan Cheng, Yimin Xiao","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-28T18:21:43Z","title":"The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6693","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:312d8824743764bfa3ba24cc094db200e99a73a0a09ef2513ba478265619c124","target":"record","created_at":"2026-05-18T01:15:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a088c45d0f63ffea4b4d2e74ce4ae07a0df4afe4e1b12247c0ca5bc48e95851d","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-28T18:21:43Z","title_canon_sha256":"93a8920d37bc209661d6b295cafa83e1d1d73deb2e4b5e83d426c7bcc9439732"},"schema_version":"1.0","source":{"id":"1211.6693","kind":"arxiv","version":3}},"canonical_sha256":"d39f5a1f02d9e336eefd0fb93c128ac6a8f1ea91c9727f1ad37311008a48f2b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d39f5a1f02d9e336eefd0fb93c128ac6a8f1ea91c9727f1ad37311008a48f2b3","first_computed_at":"2026-05-18T01:15:39.237728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:39.237728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"crOG+5uzkHk0jrXl2COtM3SweNHR//XYh+uEy95QklL6AM1s+/yjhOVIO4lg3QJ7vnkeeAaNzyXhEQldqleTDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:39.238516Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.6693","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:312d8824743764bfa3ba24cc094db200e99a73a0a09ef2513ba478265619c124","sha256:fcb614248560c33db8b1b4b39133651b4979fdfc7a1864fa338dd4bab64d6333"],"state_sha256":"72acd879f806eb831398b1fc4cf2d01508f726d27823421ef9b086c85f43cb7f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GX1Ad1vEqLtTVZvDCAF8OFhCbl/qnuAEZj5YwUf+DcQz9mLa+HLnO/sioxgJscED8iVcYzLe0yKyCdBGjCd2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T05:04:28.250221Z","bundle_sha256":"c08759d8972995131acbb8050d109e33cd2b08ef82498abb2e5c5fd03d8e72c9"}}