{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:JSXEUB6QDQFWEVRPYLM7LC5YAK","short_pith_number":"pith:JSXEUB6Q","canonical_record":{"source":{"id":"1907.07966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-18T10:23:09Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"13230351c2dcfbdcb32857bc7bc8ee2ecf217f9b5f84c88b3d5ac92829860c31","abstract_canon_sha256":"35a1310663fa29bcdc78113dc1f9bc102a3cee79ed65a4ec96edaa386cc2c691"},"schema_version":"1.0"},"canonical_sha256":"4cae4a07d01c0b62562fc2d9f58bb802a4077d11b45db40e25e1b25ff68e79b2","source":{"kind":"arxiv","id":"1907.07966","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.07966","created_at":"2026-05-17T23:40:15Z"},{"alias_kind":"arxiv_version","alias_value":"1907.07966v1","created_at":"2026-05-17T23:40:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.07966","created_at":"2026-05-17T23:40:15Z"},{"alias_kind":"pith_short_12","alias_value":"JSXEUB6QDQFW","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"JSXEUB6QDQFWEVRP","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"JSXEUB6Q","created_at":"2026-05-18T12:33:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:JSXEUB6QDQFWEVRPYLM7LC5YAK","target":"record","payload":{"canonical_record":{"source":{"id":"1907.07966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-18T10:23:09Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"13230351c2dcfbdcb32857bc7bc8ee2ecf217f9b5f84c88b3d5ac92829860c31","abstract_canon_sha256":"35a1310663fa29bcdc78113dc1f9bc102a3cee79ed65a4ec96edaa386cc2c691"},"schema_version":"1.0"},"canonical_sha256":"4cae4a07d01c0b62562fc2d9f58bb802a4077d11b45db40e25e1b25ff68e79b2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:15.546680Z","signature_b64":"Zf3MN8k126ovlfVrGvqjUyJT7vJcJb3ff84eGTdjG+moSR1PxFLhktLqj2Nw0yNP8oWbrUsM7CKA8+ZmtkdaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cae4a07d01c0b62562fc2d9f58bb802a4077d11b45db40e25e1b25ff68e79b2","last_reissued_at":"2026-05-17T23:40:15.545937Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:15.545937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.07966","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":"2026-05-17T23:40:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QRxP71yM0QoO+scLhvGB8m0/HStN2irEsTcSAgNMHh2zbNtRkH272nWFqpEIizdyhrhSygYIH/QZOqlIqBr2BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:59:26.277763Z"},"content_sha256":"98deb0d83d48f8edb9025e2c72c37dc4f50c7754c827b13802c8f9c8d5ccb233","schema_version":"1.0","event_id":"sha256:98deb0d83d48f8edb9025e2c72c37dc4f50c7754c827b13802c8f9c8d5ccb233"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:JSXEUB6QDQFWEVRPYLM7LC5YAK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bivariate Bernstein-gamma functions and moments of exponential functionals of subordinators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Adam Barker, Mladen Savov","submitted_at":"2019-07-18T10:23:09Z","abstract_excerpt":"In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions.\n  Then, we demonstrate the importance and power of these results through an application to exponential functionals of L\\'evy processes.\n  In more detail, for a subordinator (a non-decreasing L\\'evy process) $(X_s)_{s\\geq 0}$, we study its \\textit{exponential functional}, $\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07966","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":"2026-05-17T23:40:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KaqwryQDLEkaPq0Mc3GqLTqYtWloezNRd5Pkep/def+TvGad2exoW//UjKRmlOjoB+O+3MsSY9EsY1s9W/MuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:59:26.278109Z"},"content_sha256":"946f3734307bc11bfd1286fb68a06bd8d505e3a0f45f5becd7a8ca76f40a64cd","schema_version":"1.0","event_id":"sha256:946f3734307bc11bfd1286fb68a06bd8d505e3a0f45f5becd7a8ca76f40a64cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK/bundle.json","state_url":"https://pith.science/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK/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":"2026-06-03T18:59:26Z","links":{"resolver":"https://pith.science/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK","bundle":"https://pith.science/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK/bundle.json","state":"https://pith.science/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JSXEUB6QDQFWEVRPYLM7LC5YAK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:JSXEUB6QDQFWEVRPYLM7LC5YAK","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":"35a1310663fa29bcdc78113dc1f9bc102a3cee79ed65a4ec96edaa386cc2c691","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-18T10:23:09Z","title_canon_sha256":"13230351c2dcfbdcb32857bc7bc8ee2ecf217f9b5f84c88b3d5ac92829860c31"},"schema_version":"1.0","source":{"id":"1907.07966","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.07966","created_at":"2026-05-17T23:40:15Z"},{"alias_kind":"arxiv_version","alias_value":"1907.07966v1","created_at":"2026-05-17T23:40:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.07966","created_at":"2026-05-17T23:40:15Z"},{"alias_kind":"pith_short_12","alias_value":"JSXEUB6QDQFW","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"JSXEUB6QDQFWEVRP","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"JSXEUB6Q","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:946f3734307bc11bfd1286fb68a06bd8d505e3a0f45f5becd7a8ca76f40a64cd","target":"graph","created_at":"2026-05-17T23:40:15Z","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 extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions.\n  Then, we demonstrate the importance and power of these results through an application to exponential functionals of L\\'evy processes.\n  In more detail, for a subordinator (a non-decreasing L\\'evy process) $(X_s)_{s\\geq 0}$, we study its \\textit{exponential functional}, $\\in","authors_text":"Adam Barker, Mladen Savov","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-18T10:23:09Z","title":"Bivariate Bernstein-gamma functions and moments of exponential functionals of subordinators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07966","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":"sha256:98deb0d83d48f8edb9025e2c72c37dc4f50c7754c827b13802c8f9c8d5ccb233","target":"record","created_at":"2026-05-17T23:40:15Z","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":"35a1310663fa29bcdc78113dc1f9bc102a3cee79ed65a4ec96edaa386cc2c691","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-18T10:23:09Z","title_canon_sha256":"13230351c2dcfbdcb32857bc7bc8ee2ecf217f9b5f84c88b3d5ac92829860c31"},"schema_version":"1.0","source":{"id":"1907.07966","kind":"arxiv","version":1}},"canonical_sha256":"4cae4a07d01c0b62562fc2d9f58bb802a4077d11b45db40e25e1b25ff68e79b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4cae4a07d01c0b62562fc2d9f58bb802a4077d11b45db40e25e1b25ff68e79b2","first_computed_at":"2026-05-17T23:40:15.545937Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:15.545937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Zf3MN8k126ovlfVrGvqjUyJT7vJcJb3ff84eGTdjG+moSR1PxFLhktLqj2Nw0yNP8oWbrUsM7CKA8+ZmtkdaBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:15.546680Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.07966","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:98deb0d83d48f8edb9025e2c72c37dc4f50c7754c827b13802c8f9c8d5ccb233","sha256:946f3734307bc11bfd1286fb68a06bd8d505e3a0f45f5becd7a8ca76f40a64cd"],"state_sha256":"d73b560d2e8ab06bf78ec2bdd2dca313535af2bfd14c26888af40d46b1212595"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xJGPZSSgSAagZPVAKatc16q0hkU/Bo5bO+ejPdXpyrJYAhID7vjWwqjyahb+wEnngVqmdmgCUyZp6MU4ZXOeAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T18:59:26.280060Z","bundle_sha256":"018a1429d8a8b8de15a4e17d435010c14316cfe6b8e625e921a74725bb6d53c3"}}