{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZF63UVSQAHQLRGKRA6FD4ESVJL","short_pith_number":"pith:ZF63UVSQ","canonical_record":{"source":{"id":"1602.04480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-14T18:04:21Z","cross_cats_sorted":[],"title_canon_sha256":"b2123e7eed4bd2f61e1f2a3296073d069a5d52d7dd80fc8b6026de3260c6b836","abstract_canon_sha256":"7bdaffc47e0402ddf92b17ad65d4a45b2d652fc61f5b34c7acd8c62c0d148f6b"},"schema_version":"1.0"},"canonical_sha256":"c97dba565001e0b89951078a3e12554ac63cace0265f38e34c04094f90c2fdff","source":{"kind":"arxiv","id":"1602.04480","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04480","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04480v2","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04480","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"pith_short_12","alias_value":"ZF63UVSQAHQL","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZF63UVSQAHQLRGKR","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZF63UVSQ","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZF63UVSQAHQLRGKRA6FD4ESVJL","target":"record","payload":{"canonical_record":{"source":{"id":"1602.04480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-14T18:04:21Z","cross_cats_sorted":[],"title_canon_sha256":"b2123e7eed4bd2f61e1f2a3296073d069a5d52d7dd80fc8b6026de3260c6b836","abstract_canon_sha256":"7bdaffc47e0402ddf92b17ad65d4a45b2d652fc61f5b34c7acd8c62c0d148f6b"},"schema_version":"1.0"},"canonical_sha256":"c97dba565001e0b89951078a3e12554ac63cace0265f38e34c04094f90c2fdff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:52.947163Z","signature_b64":"5cE4ODYgh4B1TYfh4g86DQZQA9GLzLarD7sVnFTyJCnq0wMM95I44v3ZaqHjtcEzoroi5Jup3xhCNsv4/oDWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c97dba565001e0b89951078a3e12554ac63cace0265f38e34c04094f90c2fdff","last_reissued_at":"2026-05-18T01:19:52.946598Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:52.946598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.04480","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:19:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q+MkzKGv4wcPkItgq75GkmfgaGLuVW/YDf0l4NmI1729/3bjuOAhY0ImCeofo/VdVUolfAW0tM0VUacNdEn+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T02:41:23.430596Z"},"content_sha256":"c88486c809ef6ef58aacd8080e9fb35acd5e93b46742393bfbfbcaa4f9d5d6cb","schema_version":"1.0","event_id":"sha256:c88486c809ef6ef58aacd8080e9fb35acd5e93b46742393bfbfbcaa4f9d5d6cb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZF63UVSQAHQLRGKRA6FD4ESVJL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"From Doob's maximal identity to Azema supermartingale","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shiqi Song","submitted_at":"2016-02-14T18:04:21Z","abstract_excerpt":"We characterize the random times $\\rho$ whose Azema supermartingales $Z^\\rho$ take the form $Z^\\rho=U/U^*$ for some non negative local martingales $U$ starting from 1 vanishing at infinity, where $U^*$ denotes the running maximum process of $U$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04480","kind":"arxiv","version":2},"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":"2026-05-18T01:19:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9nsQTKla6sxJhueNZjGZl8RHwcH+zgpWFdEmw8/Q5HO8C5YvYcWPqiFzMmAnqFK8N6khhef145igNILj9hf/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T02:41:23.430943Z"},"content_sha256":"985e23afd90534adc8c4fafcbab317ac6bbf6c7e3c99f4ac54a8be77df0e6b7d","schema_version":"1.0","event_id":"sha256:985e23afd90534adc8c4fafcbab317ac6bbf6c7e3c99f4ac54a8be77df0e6b7d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL/bundle.json","state_url":"https://pith.science/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL/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":"2026-06-30T02:41:23Z","links":{"resolver":"https://pith.science/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL","bundle":"https://pith.science/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL/bundle.json","state":"https://pith.science/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZF63UVSQAHQLRGKRA6FD4ESVJL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZF63UVSQAHQLRGKRA6FD4ESVJL","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":"7bdaffc47e0402ddf92b17ad65d4a45b2d652fc61f5b34c7acd8c62c0d148f6b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-14T18:04:21Z","title_canon_sha256":"b2123e7eed4bd2f61e1f2a3296073d069a5d52d7dd80fc8b6026de3260c6b836"},"schema_version":"1.0","source":{"id":"1602.04480","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04480","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04480v2","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04480","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"pith_short_12","alias_value":"ZF63UVSQAHQL","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZF63UVSQAHQLRGKR","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZF63UVSQ","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:985e23afd90534adc8c4fafcbab317ac6bbf6c7e3c99f4ac54a8be77df0e6b7d","target":"graph","created_at":"2026-05-18T01:19:52Z","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":"We characterize the random times $\\rho$ whose Azema supermartingales $Z^\\rho$ take the form $Z^\\rho=U/U^*$ for some non negative local martingales $U$ starting from 1 vanishing at infinity, where $U^*$ denotes the running maximum process of $U$.","authors_text":"Shiqi Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-14T18:04:21Z","title":"From Doob's maximal identity to Azema supermartingale"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04480","kind":"arxiv","version":2},"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:c88486c809ef6ef58aacd8080e9fb35acd5e93b46742393bfbfbcaa4f9d5d6cb","target":"record","created_at":"2026-05-18T01:19:52Z","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":"7bdaffc47e0402ddf92b17ad65d4a45b2d652fc61f5b34c7acd8c62c0d148f6b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-14T18:04:21Z","title_canon_sha256":"b2123e7eed4bd2f61e1f2a3296073d069a5d52d7dd80fc8b6026de3260c6b836"},"schema_version":"1.0","source":{"id":"1602.04480","kind":"arxiv","version":2}},"canonical_sha256":"c97dba565001e0b89951078a3e12554ac63cace0265f38e34c04094f90c2fdff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c97dba565001e0b89951078a3e12554ac63cace0265f38e34c04094f90c2fdff","first_computed_at":"2026-05-18T01:19:52.946598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:52.946598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5cE4ODYgh4B1TYfh4g86DQZQA9GLzLarD7sVnFTyJCnq0wMM95I44v3ZaqHjtcEzoroi5Jup3xhCNsv4/oDWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:52.947163Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.04480","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c88486c809ef6ef58aacd8080e9fb35acd5e93b46742393bfbfbcaa4f9d5d6cb","sha256:985e23afd90534adc8c4fafcbab317ac6bbf6c7e3c99f4ac54a8be77df0e6b7d"],"state_sha256":"a906b560fc457c5303b065a1e129cb747664a53aa29e05a5b27b3bae738ea60f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NInhG0SUesM53jdQrVCsVJhHUfXZJEQuFgVN4+9dJQ9G4BD53wcnqlynY3zryGUKMuW00oHtFziZBAZKfxhNDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T02:41:23.432913Z","bundle_sha256":"fd83e190e430480ae81e7d5161b47b443923700e225328eac33c2ddff0a5e00d"}}