{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:AQNMX3HEAIYQ573HEZHJTF5VM2","short_pith_number":"pith:AQNMX3HE","canonical_record":{"source":{"id":"1109.4359","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T17:03:45Z","cross_cats_sorted":[],"title_canon_sha256":"4bed6660dcab96ab01ca131535bafbec21e48b84c1b64218a5b9058d34002d2f","abstract_canon_sha256":"58187864c31bf5e6f6a44f3a257acff8b3bff31476c942f1515a0b169a1ae90c"},"schema_version":"1.0"},"canonical_sha256":"041acbece402310eff67264e9997b566a38dc64e1bd3395da156cf4e7253aa38","source":{"kind":"arxiv","id":"1109.4359","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4359","created_at":"2026-05-18T03:06:52Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4359v5","created_at":"2026-05-18T03:06:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4359","created_at":"2026-05-18T03:06:52Z"},{"alias_kind":"pith_short_12","alias_value":"AQNMX3HEAIYQ","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AQNMX3HEAIYQ573H","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AQNMX3HE","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:AQNMX3HEAIYQ573HEZHJTF5VM2","target":"record","payload":{"canonical_record":{"source":{"id":"1109.4359","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T17:03:45Z","cross_cats_sorted":[],"title_canon_sha256":"4bed6660dcab96ab01ca131535bafbec21e48b84c1b64218a5b9058d34002d2f","abstract_canon_sha256":"58187864c31bf5e6f6a44f3a257acff8b3bff31476c942f1515a0b169a1ae90c"},"schema_version":"1.0"},"canonical_sha256":"041acbece402310eff67264e9997b566a38dc64e1bd3395da156cf4e7253aa38","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:52.103778Z","signature_b64":"RV1aLUaOVix8xPwPqmQKntzxoJ6hqc3Y4uKS4bNjRTKuvqRqIYv1tLQUvEL2XlnsgOtUzcZc+a74oWlPWo7pDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"041acbece402310eff67264e9997b566a38dc64e1bd3395da156cf4e7253aa38","last_reissued_at":"2026-05-18T03:06:52.103131Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:52.103131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.4359","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:06:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wTAbfYz2dk3arQmFfL+aewKIS+IWvTW/CsXJa+OTL5sacYXPwkViL4JeYKnsQw8T6iogu4wWMjnqB8jsPEAjDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T04:26:27.822256Z"},"content_sha256":"304f6f066590460d0ad47fe6f0ed2c959d8aaf80bf38caf093abef643936760a","schema_version":"1.0","event_id":"sha256:304f6f066590460d0ad47fe6f0ed2c959d8aaf80bf38caf093abef643936760a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:AQNMX3HEAIYQ573HEZHJTF5VM2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hoeffding's inequality for supermartingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ion Grama, Quansheng Liu, Xiequan Fan","submitted_at":"2011-09-20T17:03:45Z","abstract_excerpt":"We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4359","kind":"arxiv","version":5},"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":"2026-05-18T03:06:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qh/9T3sW0zgSzSfktUYWYBQrtWs2GMtfm767YU/LQ9c2cqLTvQBxnukqtpc9O56RFNAubS/DCboOJgfpu9o8DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T04:26:27.822988Z"},"content_sha256":"89cfa6b69efc947de59be078c12fd3dc7f843987bcb470da5d327c37043a6d64","schema_version":"1.0","event_id":"sha256:89cfa6b69efc947de59be078c12fd3dc7f843987bcb470da5d327c37043a6d64"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AQNMX3HEAIYQ573HEZHJTF5VM2/bundle.json","state_url":"https://pith.science/pith/AQNMX3HEAIYQ573HEZHJTF5VM2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AQNMX3HEAIYQ573HEZHJTF5VM2/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":"2026-05-31T04:26:27Z","links":{"resolver":"https://pith.science/pith/AQNMX3HEAIYQ573HEZHJTF5VM2","bundle":"https://pith.science/pith/AQNMX3HEAIYQ573HEZHJTF5VM2/bundle.json","state":"https://pith.science/pith/AQNMX3HEAIYQ573HEZHJTF5VM2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AQNMX3HEAIYQ573HEZHJTF5VM2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:AQNMX3HEAIYQ573HEZHJTF5VM2","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":"58187864c31bf5e6f6a44f3a257acff8b3bff31476c942f1515a0b169a1ae90c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T17:03:45Z","title_canon_sha256":"4bed6660dcab96ab01ca131535bafbec21e48b84c1b64218a5b9058d34002d2f"},"schema_version":"1.0","source":{"id":"1109.4359","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4359","created_at":"2026-05-18T03:06:52Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4359v5","created_at":"2026-05-18T03:06:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4359","created_at":"2026-05-18T03:06:52Z"},{"alias_kind":"pith_short_12","alias_value":"AQNMX3HEAIYQ","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AQNMX3HEAIYQ573H","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AQNMX3HE","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:89cfa6b69efc947de59be078c12fd3dc7f843987bcb470da5d327c37043a6d64","target":"graph","created_at":"2026-05-18T03:06:52Z","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 give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.","authors_text":"Ion Grama, Quansheng Liu, Xiequan Fan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T17:03:45Z","title":"Hoeffding's inequality for supermartingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4359","kind":"arxiv","version":5},"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":"sha256:304f6f066590460d0ad47fe6f0ed2c959d8aaf80bf38caf093abef643936760a","target":"record","created_at":"2026-05-18T03:06:52Z","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":"58187864c31bf5e6f6a44f3a257acff8b3bff31476c942f1515a0b169a1ae90c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T17:03:45Z","title_canon_sha256":"4bed6660dcab96ab01ca131535bafbec21e48b84c1b64218a5b9058d34002d2f"},"schema_version":"1.0","source":{"id":"1109.4359","kind":"arxiv","version":5}},"canonical_sha256":"041acbece402310eff67264e9997b566a38dc64e1bd3395da156cf4e7253aa38","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"041acbece402310eff67264e9997b566a38dc64e1bd3395da156cf4e7253aa38","first_computed_at":"2026-05-18T03:06:52.103131Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:52.103131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RV1aLUaOVix8xPwPqmQKntzxoJ6hqc3Y4uKS4bNjRTKuvqRqIYv1tLQUvEL2XlnsgOtUzcZc+a74oWlPWo7pDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:52.103778Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4359","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:304f6f066590460d0ad47fe6f0ed2c959d8aaf80bf38caf093abef643936760a","sha256:89cfa6b69efc947de59be078c12fd3dc7f843987bcb470da5d327c37043a6d64"],"state_sha256":"167102f52d387df563f0ee3a8edf1462be2685330206010f80fd6a03ff3fda22"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1wQVUKIx9BeYPV6toi2JsHdDBQsq5mB6YwoJDQn+zuy9f/CNn6zePCAwrphkjwmUPWM8l+qrVbvHDFl4F7ZgBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T04:26:27.826802Z","bundle_sha256":"c54d6f0bf0ee680e85229575795f489f2722f4a79f2d3e9c626e4297dd840e5e"}}