{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:OHCSYESHTT3LBBINGJTT22LP6M","short_pith_number":"pith:OHCSYESH","canonical_record":{"source":{"id":"1811.04567","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-12T05:59:13Z","cross_cats_sorted":[],"title_canon_sha256":"6e90b0af1992e68a27f465412b97cd8ec090b27ac18779319fa9f9fb2a4a0a47","abstract_canon_sha256":"fd54d02fbd66b13dfc675533011a87ab6b74870c340719b995c897de3e9a8011"},"schema_version":"1.0"},"canonical_sha256":"71c52c12479cf6b0850d32673d696ff3174dfae2bb4a8b83f1788b6ebf9b8af3","source":{"kind":"arxiv","id":"1811.04567","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.04567","created_at":"2026-05-18T00:01:04Z"},{"alias_kind":"arxiv_version","alias_value":"1811.04567v1","created_at":"2026-05-18T00:01:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.04567","created_at":"2026-05-18T00:01:04Z"},{"alias_kind":"pith_short_12","alias_value":"OHCSYESHTT3L","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OHCSYESHTT3LBBIN","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OHCSYESH","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:OHCSYESHTT3LBBINGJTT22LP6M","target":"record","payload":{"canonical_record":{"source":{"id":"1811.04567","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-12T05:59:13Z","cross_cats_sorted":[],"title_canon_sha256":"6e90b0af1992e68a27f465412b97cd8ec090b27ac18779319fa9f9fb2a4a0a47","abstract_canon_sha256":"fd54d02fbd66b13dfc675533011a87ab6b74870c340719b995c897de3e9a8011"},"schema_version":"1.0"},"canonical_sha256":"71c52c12479cf6b0850d32673d696ff3174dfae2bb4a8b83f1788b6ebf9b8af3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:04.756648Z","signature_b64":"tEYr5+L8SSJtBqrWT+Vn/2K72URKvbIgZbu1Jj8y1eV2K7ojSHKXT5D/C0NHjf+LNk4ILvsNzXX1xILwTBAeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71c52c12479cf6b0850d32673d696ff3174dfae2bb4a8b83f1788b6ebf9b8af3","last_reissued_at":"2026-05-18T00:01:04.756026Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:04.756026Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.04567","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":"2026-05-18T00:01:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dnE/pR+jwZPiefntsaNKzgKEC9EYzGpmWucqLpOQosx/1hiV6REQfyYvzEOpcsus/oeKGMWCQ3qlirRZr/5bAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T14:37:32.099273Z"},"content_sha256":"d0958d8dd38c0e4ef89b5758eaa6271226a39cf7a4fc062b3c4a497cacd6d5f6","schema_version":"1.0","event_id":"sha256:d0958d8dd38c0e4ef89b5758eaa6271226a39cf7a4fc062b3c4a497cacd6d5f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:OHCSYESHTT3LBBINGJTT22LP6M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Time-changed Poisson processes of order $k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. Maheshwari, Ayushi S. Sengar, N. S. Upadhye","submitted_at":"2018-11-12T05:59:13Z","abstract_excerpt":"In this article, we study the Poisson process of order k (PPoK) time-changed with an independent L\\'evy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.04567","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":"2026-05-18T00:01:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CUHzH2n4zD4hITRIHRKFh8otVn1f6iXRQJUsnvOtPy8XkowflilS2mOxQOl+pblvUeV32UXLti3THqzy5eLbAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T14:37:32.099660Z"},"content_sha256":"3e5c4853c125c292b130129696992df40695486a87d8ff12cd54ec38bb923674","schema_version":"1.0","event_id":"sha256:3e5c4853c125c292b130129696992df40695486a87d8ff12cd54ec38bb923674"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OHCSYESHTT3LBBINGJTT22LP6M/bundle.json","state_url":"https://pith.science/pith/OHCSYESHTT3LBBINGJTT22LP6M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OHCSYESHTT3LBBINGJTT22LP6M/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":"2026-06-24T14:37:32Z","links":{"resolver":"https://pith.science/pith/OHCSYESHTT3LBBINGJTT22LP6M","bundle":"https://pith.science/pith/OHCSYESHTT3LBBINGJTT22LP6M/bundle.json","state":"https://pith.science/pith/OHCSYESHTT3LBBINGJTT22LP6M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OHCSYESHTT3LBBINGJTT22LP6M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:OHCSYESHTT3LBBINGJTT22LP6M","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":"fd54d02fbd66b13dfc675533011a87ab6b74870c340719b995c897de3e9a8011","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-12T05:59:13Z","title_canon_sha256":"6e90b0af1992e68a27f465412b97cd8ec090b27ac18779319fa9f9fb2a4a0a47"},"schema_version":"1.0","source":{"id":"1811.04567","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.04567","created_at":"2026-05-18T00:01:04Z"},{"alias_kind":"arxiv_version","alias_value":"1811.04567v1","created_at":"2026-05-18T00:01:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.04567","created_at":"2026-05-18T00:01:04Z"},{"alias_kind":"pith_short_12","alias_value":"OHCSYESHTT3L","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OHCSYESHTT3LBBIN","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OHCSYESH","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:3e5c4853c125c292b130129696992df40695486a87d8ff12cd54ec38bb923674","target":"graph","created_at":"2026-05-18T00:01:04Z","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 article, we study the Poisson process of order k (PPoK) time-changed with an independent L\\'evy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, w","authors_text":"A. Maheshwari, Ayushi S. Sengar, N. S. Upadhye","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-12T05:59:13Z","title":"Time-changed Poisson processes of order $k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.04567","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:d0958d8dd38c0e4ef89b5758eaa6271226a39cf7a4fc062b3c4a497cacd6d5f6","target":"record","created_at":"2026-05-18T00:01:04Z","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":"fd54d02fbd66b13dfc675533011a87ab6b74870c340719b995c897de3e9a8011","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-12T05:59:13Z","title_canon_sha256":"6e90b0af1992e68a27f465412b97cd8ec090b27ac18779319fa9f9fb2a4a0a47"},"schema_version":"1.0","source":{"id":"1811.04567","kind":"arxiv","version":1}},"canonical_sha256":"71c52c12479cf6b0850d32673d696ff3174dfae2bb4a8b83f1788b6ebf9b8af3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71c52c12479cf6b0850d32673d696ff3174dfae2bb4a8b83f1788b6ebf9b8af3","first_computed_at":"2026-05-18T00:01:04.756026Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:04.756026Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tEYr5+L8SSJtBqrWT+Vn/2K72URKvbIgZbu1Jj8y1eV2K7ojSHKXT5D/C0NHjf+LNk4ILvsNzXX1xILwTBAeDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:04.756648Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.04567","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0958d8dd38c0e4ef89b5758eaa6271226a39cf7a4fc062b3c4a497cacd6d5f6","sha256:3e5c4853c125c292b130129696992df40695486a87d8ff12cd54ec38bb923674"],"state_sha256":"b4b28fb61fc3ce0ad55467dcd2c502a5781c6c0f00b67cce30577dedd887441a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fr0Zk7ikVzrOsXVPNbnuYjA9Pi5UknyJQs9ZbJFlQNR23SH1gpyNt2VMVcvEphcgKcSigppGPqkAqUSc/gZsAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T14:37:32.101584Z","bundle_sha256":"77cb646bc9d8c215ca24ab7789dd59806de8a1d710b1297656784ca73a7e9ef3"}}