{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:2LQKFVSSULDINLUAKJKXHIJWNP","short_pith_number":"pith:2LQKFVSS","canonical_record":{"source":{"id":"1803.09985","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-27T09:38:48Z","cross_cats_sorted":[],"title_canon_sha256":"6bcdda2fe6a7129f38ec393c9c27a70940cd5715ff8f0e3d184aaaa3af3adf22","abstract_canon_sha256":"07cd427f58ec4f099f529a04c3003d23a6fd991dce96daaec682906297fd1c9d"},"schema_version":"1.0"},"canonical_sha256":"d2e0a2d652a2c686ae80525573a1366bd15dd5e5371c38682bffa5aa2b5f4367","source":{"kind":"arxiv","id":"1803.09985","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.09985","created_at":"2026-05-18T00:20:00Z"},{"alias_kind":"arxiv_version","alias_value":"1803.09985v1","created_at":"2026-05-18T00:20:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09985","created_at":"2026-05-18T00:20:00Z"},{"alias_kind":"pith_short_12","alias_value":"2LQKFVSSULDI","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2LQKFVSSULDINLUA","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2LQKFVSS","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:2LQKFVSSULDINLUAKJKXHIJWNP","target":"record","payload":{"canonical_record":{"source":{"id":"1803.09985","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-27T09:38:48Z","cross_cats_sorted":[],"title_canon_sha256":"6bcdda2fe6a7129f38ec393c9c27a70940cd5715ff8f0e3d184aaaa3af3adf22","abstract_canon_sha256":"07cd427f58ec4f099f529a04c3003d23a6fd991dce96daaec682906297fd1c9d"},"schema_version":"1.0"},"canonical_sha256":"d2e0a2d652a2c686ae80525573a1366bd15dd5e5371c38682bffa5aa2b5f4367","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:00.806710Z","signature_b64":"kTIZoSqP0Li9At4snT5oCeepTaFEcJmnnGHd+aOD689pKtibhStKMiUnnxunnY5k19JzAQVSuf1mHXrl4avGBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2e0a2d652a2c686ae80525573a1366bd15dd5e5371c38682bffa5aa2b5f4367","last_reissued_at":"2026-05-18T00:20:00.806105Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:00.806105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.09985","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:20:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4R3Bg4Br9UaJcdmu6wLCTrkYeXe1cuBpby5v6v7LwTQGzmhQOWckdkULmWDw9Zo2eDxtCjsL25NvSbJCCAXYCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T21:27:49.586170Z"},"content_sha256":"b17586a20f4e3f3bb21ba3404937b4ae76af6ecac8ebb68c32755af7fc3b1382","schema_version":"1.0","event_id":"sha256:b17586a20f4e3f3bb21ba3404937b4ae76af6ecac8ebb68c32755af7fc3b1382"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:2LQKFVSSULDINLUAKJKXHIJWNP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New contributions to the study of stochastic processes of the class $(\\Sigma)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fulgence Eyi Obiang, Octave Moutsinga, Youssef Youssef","submitted_at":"2018-03-27T09:38:48Z","abstract_excerpt":"In this paper, we contribute to the study of the class $(\\Sigma)$. In the first part of the paper, we provide new ways to characterize stochastic processes of the above mentioned class and we derive some new properties. For instance, we prove that a stochastic process $X$ is an element of the class $(\\Sigma)$ if, and only if, its absolute value is equal to absolute value of some martingale $M$. In the second part, we study in particular, stochastic processes of the class $(\\Sigma)$ which vanish on the zero set of a given Brownian motion. More precisely, we provide a characterization theorem an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09985","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:20:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UoP010qvJW7b6A9fv7KcOcaaYLEkICS3xuvFCfG3Jj+3jRqfCgJh5qTv6PcLHvGS8Px0iN/Fh369g4gkifzfCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T21:27:49.586514Z"},"content_sha256":"dbc8a52de4268d633a6b45c7fe0440a32c6916703cbd7914642de0645d49618c","schema_version":"1.0","event_id":"sha256:dbc8a52de4268d633a6b45c7fe0440a32c6916703cbd7914642de0645d49618c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2LQKFVSSULDINLUAKJKXHIJWNP/bundle.json","state_url":"https://pith.science/pith/2LQKFVSSULDINLUAKJKXHIJWNP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2LQKFVSSULDINLUAKJKXHIJWNP/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":"2026-06-29T21:27:49Z","links":{"resolver":"https://pith.science/pith/2LQKFVSSULDINLUAKJKXHIJWNP","bundle":"https://pith.science/pith/2LQKFVSSULDINLUAKJKXHIJWNP/bundle.json","state":"https://pith.science/pith/2LQKFVSSULDINLUAKJKXHIJWNP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2LQKFVSSULDINLUAKJKXHIJWNP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:2LQKFVSSULDINLUAKJKXHIJWNP","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":"07cd427f58ec4f099f529a04c3003d23a6fd991dce96daaec682906297fd1c9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-27T09:38:48Z","title_canon_sha256":"6bcdda2fe6a7129f38ec393c9c27a70940cd5715ff8f0e3d184aaaa3af3adf22"},"schema_version":"1.0","source":{"id":"1803.09985","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.09985","created_at":"2026-05-18T00:20:00Z"},{"alias_kind":"arxiv_version","alias_value":"1803.09985v1","created_at":"2026-05-18T00:20:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09985","created_at":"2026-05-18T00:20:00Z"},{"alias_kind":"pith_short_12","alias_value":"2LQKFVSSULDI","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2LQKFVSSULDINLUA","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2LQKFVSS","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:dbc8a52de4268d633a6b45c7fe0440a32c6916703cbd7914642de0645d49618c","target":"graph","created_at":"2026-05-18T00:20:00Z","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 contribute to the study of the class $(\\Sigma)$. In the first part of the paper, we provide new ways to characterize stochastic processes of the above mentioned class and we derive some new properties. For instance, we prove that a stochastic process $X$ is an element of the class $(\\Sigma)$ if, and only if, its absolute value is equal to absolute value of some martingale $M$. In the second part, we study in particular, stochastic processes of the class $(\\Sigma)$ which vanish on the zero set of a given Brownian motion. More precisely, we provide a characterization theorem an","authors_text":"Fulgence Eyi Obiang, Octave Moutsinga, Youssef Youssef","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-27T09:38:48Z","title":"New contributions to the study of stochastic processes of the class $(\\Sigma)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09985","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:b17586a20f4e3f3bb21ba3404937b4ae76af6ecac8ebb68c32755af7fc3b1382","target":"record","created_at":"2026-05-18T00:20:00Z","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":"07cd427f58ec4f099f529a04c3003d23a6fd991dce96daaec682906297fd1c9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-27T09:38:48Z","title_canon_sha256":"6bcdda2fe6a7129f38ec393c9c27a70940cd5715ff8f0e3d184aaaa3af3adf22"},"schema_version":"1.0","source":{"id":"1803.09985","kind":"arxiv","version":1}},"canonical_sha256":"d2e0a2d652a2c686ae80525573a1366bd15dd5e5371c38682bffa5aa2b5f4367","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2e0a2d652a2c686ae80525573a1366bd15dd5e5371c38682bffa5aa2b5f4367","first_computed_at":"2026-05-18T00:20:00.806105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:00.806105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kTIZoSqP0Li9At4snT5oCeepTaFEcJmnnGHd+aOD689pKtibhStKMiUnnxunnY5k19JzAQVSuf1mHXrl4avGBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:00.806710Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.09985","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b17586a20f4e3f3bb21ba3404937b4ae76af6ecac8ebb68c32755af7fc3b1382","sha256:dbc8a52de4268d633a6b45c7fe0440a32c6916703cbd7914642de0645d49618c"],"state_sha256":"1ee98c13e264b0495cff698678e6156d533f24d42f2ce5d6872d0596d6fe6119"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2BYu9CEUwwRimXwwrvWrFK4nQs9TImJ+GuEooCGcRCAhvXky06gmGhsz3z7msZk7jEjk5JkHH9SmBPpWnLElAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T21:27:49.588390Z","bundle_sha256":"3e17db81e1eb3f8d643c323ed6f5e5a810374b95ffa0983130b787011b1b52b4"}}