{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ENBK4745T3PWCOUG64WYQ2QJ72","short_pith_number":"pith:ENBK4745","canonical_record":{"source":{"id":"1410.6731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-24T16:24:54Z","cross_cats_sorted":[],"title_canon_sha256":"3d0c18f0351aea91330443ae43c7b31100d5b107654f0309784cbc5563472e2b","abstract_canon_sha256":"1cece3a38c4f8f4ff3a1c93f3f92494043fca246a94967117f48fd3df4cc8d1c"},"schema_version":"1.0"},"canonical_sha256":"2342ae7f9d9edf613a86f72d886a09fe952531ed3f67e79d2d3ad3554b3662f1","source":{"kind":"arxiv","id":"1410.6731","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.6731","created_at":"2026-05-18T00:47:29Z"},{"alias_kind":"arxiv_version","alias_value":"1410.6731v3","created_at":"2026-05-18T00:47:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.6731","created_at":"2026-05-18T00:47:29Z"},{"alias_kind":"pith_short_12","alias_value":"ENBK4745T3PW","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"ENBK4745T3PWCOUG","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"ENBK4745","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ENBK4745T3PWCOUG64WYQ2QJ72","target":"record","payload":{"canonical_record":{"source":{"id":"1410.6731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-24T16:24:54Z","cross_cats_sorted":[],"title_canon_sha256":"3d0c18f0351aea91330443ae43c7b31100d5b107654f0309784cbc5563472e2b","abstract_canon_sha256":"1cece3a38c4f8f4ff3a1c93f3f92494043fca246a94967117f48fd3df4cc8d1c"},"schema_version":"1.0"},"canonical_sha256":"2342ae7f9d9edf613a86f72d886a09fe952531ed3f67e79d2d3ad3554b3662f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:29.743154Z","signature_b64":"Ejriu0KTpbqcq0M80COErE0Fyflf9FhkTCi1u1gaOSo/KjRpk0B7q567O5p7QCTKrfqkxpTuXKAZoqhP+qOYBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2342ae7f9d9edf613a86f72d886a09fe952531ed3f67e79d2d3ad3554b3662f1","last_reissued_at":"2026-05-18T00:47:29.742706Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:29.742706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.6731","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":"2026-05-18T00:47:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JgWOGo8yuXB73PusH3RK7UsqtmhESJT5Aqd1eLr+HbXN0ZPt8UorOoCZI4K4XGjBXpIbNkjivSjA5T9WQj2LAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:52:50.627400Z"},"content_sha256":"293a4928f81228098f0697619ce955cb7a73fa6857632525690eadd77e29f8e6","schema_version":"1.0","event_id":"sha256:293a4928f81228098f0697619ce955cb7a73fa6857632525690eadd77e29f8e6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ENBK4745T3PWCOUG64WYQ2QJ72","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Markov processes, polynomial martingales and orthogonal polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pawe{\\l} J. Szab{\\l}owski","submitted_at":"2014-10-24T16:24:54Z","abstract_excerpt":"We study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It turns out that then there exists a family of polynomial martingales $\\left\\{ M_{n}(X_{t},t)\\right\\}_{n\\geq1}$ that contains complete information on the distribution (both marginal and transitional) of the process.\n  We specify conditions expressed in terms of $M_{n}^{\\prime}s$ under which a given process has independent increments and further is a Levy process, contains reversed martingales, is a harness or quadratic ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6731","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":"2026-05-18T00:47:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pg6SSQgOv+Ak08UA8EmtX1De6PUNnQroIhTW3uvrs2vD1o+0nYMWp7hAo/ahviWXmSvrX2jlPziD/YMqM8zcAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:52:50.627743Z"},"content_sha256":"c1837a9d4221c21fc756864f908b8130497cced6991a98c6970c1b0a1dd06fa3","schema_version":"1.0","event_id":"sha256:c1837a9d4221c21fc756864f908b8130497cced6991a98c6970c1b0a1dd06fa3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ENBK4745T3PWCOUG64WYQ2QJ72/bundle.json","state_url":"https://pith.science/pith/ENBK4745T3PWCOUG64WYQ2QJ72/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ENBK4745T3PWCOUG64WYQ2QJ72/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":"2026-06-25T12:52:50Z","links":{"resolver":"https://pith.science/pith/ENBK4745T3PWCOUG64WYQ2QJ72","bundle":"https://pith.science/pith/ENBK4745T3PWCOUG64WYQ2QJ72/bundle.json","state":"https://pith.science/pith/ENBK4745T3PWCOUG64WYQ2QJ72/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ENBK4745T3PWCOUG64WYQ2QJ72/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ENBK4745T3PWCOUG64WYQ2QJ72","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":"1cece3a38c4f8f4ff3a1c93f3f92494043fca246a94967117f48fd3df4cc8d1c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-24T16:24:54Z","title_canon_sha256":"3d0c18f0351aea91330443ae43c7b31100d5b107654f0309784cbc5563472e2b"},"schema_version":"1.0","source":{"id":"1410.6731","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.6731","created_at":"2026-05-18T00:47:29Z"},{"alias_kind":"arxiv_version","alias_value":"1410.6731v3","created_at":"2026-05-18T00:47:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.6731","created_at":"2026-05-18T00:47:29Z"},{"alias_kind":"pith_short_12","alias_value":"ENBK4745T3PW","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"ENBK4745T3PWCOUG","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"ENBK4745","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:c1837a9d4221c21fc756864f908b8130497cced6991a98c6970c1b0a1dd06fa3","target":"graph","created_at":"2026-05-18T00:47:29Z","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 study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It turns out that then there exists a family of polynomial martingales $\\left\\{ M_{n}(X_{t},t)\\right\\}_{n\\geq1}$ that contains complete information on the distribution (both marginal and transitional) of the process.\n  We specify conditions expressed in terms of $M_{n}^{\\prime}s$ under which a given process has independent increments and further is a Levy process, contains reversed martingales, is a harness or quadratic ha","authors_text":"Pawe{\\l} J. Szab{\\l}owski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-24T16:24:54Z","title":"Markov processes, polynomial martingales and orthogonal polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6731","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":"sha256:293a4928f81228098f0697619ce955cb7a73fa6857632525690eadd77e29f8e6","target":"record","created_at":"2026-05-18T00:47:29Z","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":"1cece3a38c4f8f4ff3a1c93f3f92494043fca246a94967117f48fd3df4cc8d1c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-24T16:24:54Z","title_canon_sha256":"3d0c18f0351aea91330443ae43c7b31100d5b107654f0309784cbc5563472e2b"},"schema_version":"1.0","source":{"id":"1410.6731","kind":"arxiv","version":3}},"canonical_sha256":"2342ae7f9d9edf613a86f72d886a09fe952531ed3f67e79d2d3ad3554b3662f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2342ae7f9d9edf613a86f72d886a09fe952531ed3f67e79d2d3ad3554b3662f1","first_computed_at":"2026-05-18T00:47:29.742706Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:47:29.742706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ejriu0KTpbqcq0M80COErE0Fyflf9FhkTCi1u1gaOSo/KjRpk0B7q567O5p7QCTKrfqkxpTuXKAZoqhP+qOYBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:47:29.743154Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.6731","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:293a4928f81228098f0697619ce955cb7a73fa6857632525690eadd77e29f8e6","sha256:c1837a9d4221c21fc756864f908b8130497cced6991a98c6970c1b0a1dd06fa3"],"state_sha256":"df9362535f4e8f100fed6a04f841242fff136c75b27b128e8bc7632056eb5baa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iS4mEl6AZidjRPKS6unN42hVY1EgnhEd0b417FJGPxCXCeIhBkwWexwIgH0c7QAtBjM9v7jbI4vGOep0sam5BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T12:52:50.629609Z","bundle_sha256":"9ab60b8e30b308ce89064aa5fdceb04eae7b7cf41230c71ba02c2686ac4ca8ab"}}