{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FPXHC6Y25XZV2YUXA2OOZ2VFZV","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":"7409dc09eac4a86c71f957fa7a2d898fcf69ef93271040027918895a875165c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-21T04:26:25Z","title_canon_sha256":"3477abcd9800096298c9e48ca0197aae0b7956c3865ae51096a3345dc27b88a3"},"schema_version":"1.0","source":{"id":"1601.05507","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05507","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05507v1","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05507","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"pith_short_12","alias_value":"FPXHC6Y25XZV","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FPXHC6Y25XZV2YUX","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FPXHC6Y2","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:bfd34c3a30a4981b732d4c825c4e29d5a42f886b937ce6e14c37f5dabcd14a61","target":"graph","created_at":"2026-05-18T01:22:13Z","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":"In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments.","authors_text":"Guomin Liu, Mingshang Hu, Qiang Gao, Xiaojun Ji","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-21T04:26:25Z","title":"Product space for two processes with independent increments under nonlinear expectations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05507","kind":"arxiv","version":1},"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:c222d17c5847cf1af018e3c066eef252d9f3375ed6245d479d84c477b7107f27","target":"record","created_at":"2026-05-18T01:22:13Z","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":"7409dc09eac4a86c71f957fa7a2d898fcf69ef93271040027918895a875165c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-21T04:26:25Z","title_canon_sha256":"3477abcd9800096298c9e48ca0197aae0b7956c3865ae51096a3345dc27b88a3"},"schema_version":"1.0","source":{"id":"1601.05507","kind":"arxiv","version":1}},"canonical_sha256":"2bee717b1aedf35d6297069ceceaa5cd613edb16ad453cd15db10179ca270dd2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bee717b1aedf35d6297069ceceaa5cd613edb16ad453cd15db10179ca270dd2","first_computed_at":"2026-05-18T01:22:13.655160Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:13.655160Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5pc1uSjYlXywtE+KnQwDdLNo64PlYv00eArxALh2fKCEXldN1TNzNEFpM8zMAFG6DzJ//AB2JeynX9MsCPZmBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:13.655720Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.05507","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c222d17c5847cf1af018e3c066eef252d9f3375ed6245d479d84c477b7107f27","sha256:bfd34c3a30a4981b732d4c825c4e29d5a42f886b937ce6e14c37f5dabcd14a61"],"state_sha256":"3ff1cb232f8f174fd86131e41bf6f92738daf9babb2328f420e9b02cd6d01126"}