{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:HFYBEV6EMTQBIIG3YGZIKBLUUQ","short_pith_number":"pith:HFYBEV6E","canonical_record":{"source":{"id":"2405.11033","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-17T18:07:48Z","cross_cats_sorted":[],"title_canon_sha256":"fd9b439be3bf32489d9764dfcf1b0ce15bb829efbf32e454622f99df3e3406b9","abstract_canon_sha256":"e4e88e0f4d7cdb6b4a0ef9b937ee0280362f6f5764bfb06f8d1cfde6fb73d8ff"},"schema_version":"1.0"},"canonical_sha256":"39701257c464e01420dbc1b2850574a400e5e0a888e409688713877c3ea2908e","source":{"kind":"arxiv","id":"2405.11033","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.11033","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"2405.11033v1","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.11033","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"HFYBEV6EMTQB","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"HFYBEV6EMTQBIIG3","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"HFYBEV6E","created_at":"2026-07-05T08:20:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:HFYBEV6EMTQBIIG3YGZIKBLUUQ","target":"record","payload":{"canonical_record":{"source":{"id":"2405.11033","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-17T18:07:48Z","cross_cats_sorted":[],"title_canon_sha256":"fd9b439be3bf32489d9764dfcf1b0ce15bb829efbf32e454622f99df3e3406b9","abstract_canon_sha256":"e4e88e0f4d7cdb6b4a0ef9b937ee0280362f6f5764bfb06f8d1cfde6fb73d8ff"},"schema_version":"1.0"},"canonical_sha256":"39701257c464e01420dbc1b2850574a400e5e0a888e409688713877c3ea2908e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:20:32.048672Z","signature_b64":"cjn96W7Qgd0/sZnX85WoR7UXxLSgQ4J+6I70oSMFlDnbk2QRY5X3FmxT72M2r7Vz4Jc8iW2UXMTc6YO1oR37BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39701257c464e01420dbc1b2850574a400e5e0a888e409688713877c3ea2908e","last_reissued_at":"2026-07-05T08:20:32.048178Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:20:32.048178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2405.11033","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":"2026-07-05T08:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"37+r4xv46o+ZmdwM55dBFfns+w9p2sM/DpqzdIpnXTLp9WhgSk1ujCbXF17lD3aqyeyJxFzUDKEM+4GS16ZeCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T08:28:47.134719Z"},"content_sha256":"2efcab4fad06a1f7f458eb15adce28d75eb61356cc342e821c7d6260ce2de4d3","schema_version":"1.0","event_id":"sha256:2efcab4fad06a1f7f458eb15adce28d75eb61356cc342e821c7d6260ce2de4d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:HFYBEV6EMTQBIIG3YGZIKBLUUQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Fractional Risk Process","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ashok Kumar Pathak, Ritik Soni","submitted_at":"2024-05-17T18:07:48Z","abstract_excerpt":"In this paper, we define a compound generalized fractional counting process (CGFCP) which is a generalization of the compound versions of several well-known fractional counting processes. We obtain its mean, variance, and the fractional differential equation governing the probability law. Motivated by Kumar et al. (2020), we introduce a fractional risk process by considering CGFCP as the surplus process and call it generalized fractional risk process (GFRP). We study the martingale property of the GFRP and show that GFRP and the associated increment process exhibit the long-range dependence (L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.11033","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.11033/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TbHvMwb0q8TAncueLHTfgLjnoT+5MDGl/mi9lIy4ofCNq+q1iCIOazi57a5Ry5jK25oaaIsThCYpicQlYFJRBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T08:28:47.135228Z"},"content_sha256":"130c997b8b774e28ae5c8f4f0e9cc307b6792fa21fe848feef54ba1264f82539","schema_version":"1.0","event_id":"sha256:130c997b8b774e28ae5c8f4f0e9cc307b6792fa21fe848feef54ba1264f82539"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ/bundle.json","state_url":"https://pith.science/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ/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":"2026-08-22T08:28:47Z","links":{"resolver":"https://pith.science/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ","bundle":"https://pith.science/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ/bundle.json","state":"https://pith.science/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HFYBEV6EMTQBIIG3YGZIKBLUUQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:HFYBEV6EMTQBIIG3YGZIKBLUUQ","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":"e4e88e0f4d7cdb6b4a0ef9b937ee0280362f6f5764bfb06f8d1cfde6fb73d8ff","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-17T18:07:48Z","title_canon_sha256":"fd9b439be3bf32489d9764dfcf1b0ce15bb829efbf32e454622f99df3e3406b9"},"schema_version":"1.0","source":{"id":"2405.11033","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.11033","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"2405.11033v1","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.11033","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"HFYBEV6EMTQB","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"HFYBEV6EMTQBIIG3","created_at":"2026-07-05T08:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"HFYBEV6E","created_at":"2026-07-05T08:20:32Z"}],"graph_snapshots":[{"event_id":"sha256:130c997b8b774e28ae5c8f4f0e9cc307b6792fa21fe848feef54ba1264f82539","target":"graph","created_at":"2026-07-05T08:20:32Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2405.11033/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we define a compound generalized fractional counting process (CGFCP) which is a generalization of the compound versions of several well-known fractional counting processes. We obtain its mean, variance, and the fractional differential equation governing the probability law. Motivated by Kumar et al. (2020), we introduce a fractional risk process by considering CGFCP as the surplus process and call it generalized fractional risk process (GFRP). We study the martingale property of the GFRP and show that GFRP and the associated increment process exhibit the long-range dependence (L","authors_text":"Ashok Kumar Pathak, Ritik Soni","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-17T18:07:48Z","title":"Generalized Fractional Risk Process"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.11033","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":"sha256:2efcab4fad06a1f7f458eb15adce28d75eb61356cc342e821c7d6260ce2de4d3","target":"record","created_at":"2026-07-05T08:20:32Z","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":"e4e88e0f4d7cdb6b4a0ef9b937ee0280362f6f5764bfb06f8d1cfde6fb73d8ff","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-17T18:07:48Z","title_canon_sha256":"fd9b439be3bf32489d9764dfcf1b0ce15bb829efbf32e454622f99df3e3406b9"},"schema_version":"1.0","source":{"id":"2405.11033","kind":"arxiv","version":1}},"canonical_sha256":"39701257c464e01420dbc1b2850574a400e5e0a888e409688713877c3ea2908e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"39701257c464e01420dbc1b2850574a400e5e0a888e409688713877c3ea2908e","first_computed_at":"2026-07-05T08:20:32.048178Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:20:32.048178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cjn96W7Qgd0/sZnX85WoR7UXxLSgQ4J+6I70oSMFlDnbk2QRY5X3FmxT72M2r7Vz4Jc8iW2UXMTc6YO1oR37BA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:20:32.048672Z","signed_message":"canonical_sha256_bytes"},"source_id":"2405.11033","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2efcab4fad06a1f7f458eb15adce28d75eb61356cc342e821c7d6260ce2de4d3","sha256:130c997b8b774e28ae5c8f4f0e9cc307b6792fa21fe848feef54ba1264f82539"],"state_sha256":"1e0f02f740eba488803f95d680d0820d5fec3069bff00d8f98ef1b930e280e3e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b1AtVeJ6fbCZ2mq07W7+tyFA4h9czO3G2HZmIgBks9t0/G7e14vsBjenSUhUYpQ096DxnD8cTkfkghhhiUWRDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T08:28:47.139835Z","bundle_sha256":"7627b7840c24d5dfae654f52dfa4079abd525425856645c4efa26bfd76e5ac30"}}