{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:P2O2OBCROKFRHLL2LE5G3SIYFH","short_pith_number":"pith:P2O2OBCR","canonical_record":{"source":{"id":"1507.06871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-24T15:01:48Z","cross_cats_sorted":[],"title_canon_sha256":"1de7bcd57bf5990efdaf3b941939378d2beacab1df82f867bf5a7e0687badb93","abstract_canon_sha256":"0770973cc5f7af844d66b0a43ebcc6bc5aa9783aa869bf1a3d530f5ba732dda5"},"schema_version":"1.0"},"canonical_sha256":"7e9da70451728b13ad7a593a6dc91829eda7b0ce36258c5a414caa00664e8296","source":{"kind":"arxiv","id":"1507.06871","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.06871","created_at":"2026-05-18T01:36:21Z"},{"alias_kind":"arxiv_version","alias_value":"1507.06871v1","created_at":"2026-05-18T01:36:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06871","created_at":"2026-05-18T01:36:21Z"},{"alias_kind":"pith_short_12","alias_value":"P2O2OBCROKFR","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"P2O2OBCROKFRHLL2","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"P2O2OBCR","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:P2O2OBCROKFRHLL2LE5G3SIYFH","target":"record","payload":{"canonical_record":{"source":{"id":"1507.06871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-24T15:01:48Z","cross_cats_sorted":[],"title_canon_sha256":"1de7bcd57bf5990efdaf3b941939378d2beacab1df82f867bf5a7e0687badb93","abstract_canon_sha256":"0770973cc5f7af844d66b0a43ebcc6bc5aa9783aa869bf1a3d530f5ba732dda5"},"schema_version":"1.0"},"canonical_sha256":"7e9da70451728b13ad7a593a6dc91829eda7b0ce36258c5a414caa00664e8296","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:21.814263Z","signature_b64":"gndL31+xKKNL7xGDwt9Xc8M5fKzuqCmEJBgwOp9ev4SpLq7cAaPlk9pCV6hUuTmiwoHxb2IXixZxOgfKu1CAAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e9da70451728b13ad7a593a6dc91829eda7b0ce36258c5a414caa00664e8296","last_reissued_at":"2026-05-18T01:36:21.813834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:21.813834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.06871","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":"2026-05-18T01:36:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R46xsY1Q8mdqd+4uhS7jhvUOlPLoC9ewV3LNuwVr2yu6mFK9zPtxSFNRM6ntn26KsxYGk3vPw/6PZJLnRU1yDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:52:46.284266Z"},"content_sha256":"db7c4708d16b858ad06906cd69f50e7ade32f7be0065ee8dd6705fae50b2aac5","schema_version":"1.0","event_id":"sha256:db7c4708d16b858ad06906cd69f50e7ade32f7be0065ee8dd6705fae50b2aac5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:P2O2OBCROKFRHLL2LE5G3SIYFH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hoeffding's inequality for sums of weakly dependent random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christos Pelekis, Jan Ramon","submitted_at":"2015-07-24T15:01:48Z","abstract_excerpt":"We provide a systematic approach to deal with the following problem. Let $X_1,\\ldots,X_n$ be, possibly dependent, $[0,1]$-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper we consider analogues of Hoeffding's result for sums of dependent random variables for which we have certain information on their dependency structure. We prove a result that yields concentration inequalitie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06871","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":"2026-05-18T01:36:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BZz75iFD4sYpFPsqlPTNJKqxoXpBFffP7zCxDFih7cqmPokMMLRaNczhM2ENMi8zkeUq/Xkwi699BjD36WBuAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:52:46.285021Z"},"content_sha256":"c6049f85a2d86abc9730fd6abb9bd9ee9bd5429be88eee742ccea2c88b296ce4","schema_version":"1.0","event_id":"sha256:c6049f85a2d86abc9730fd6abb9bd9ee9bd5429be88eee742ccea2c88b296ce4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P2O2OBCROKFRHLL2LE5G3SIYFH/bundle.json","state_url":"https://pith.science/pith/P2O2OBCROKFRHLL2LE5G3SIYFH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P2O2OBCROKFRHLL2LE5G3SIYFH/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":"2026-06-12T00:52:46Z","links":{"resolver":"https://pith.science/pith/P2O2OBCROKFRHLL2LE5G3SIYFH","bundle":"https://pith.science/pith/P2O2OBCROKFRHLL2LE5G3SIYFH/bundle.json","state":"https://pith.science/pith/P2O2OBCROKFRHLL2LE5G3SIYFH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P2O2OBCROKFRHLL2LE5G3SIYFH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:P2O2OBCROKFRHLL2LE5G3SIYFH","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":"0770973cc5f7af844d66b0a43ebcc6bc5aa9783aa869bf1a3d530f5ba732dda5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-24T15:01:48Z","title_canon_sha256":"1de7bcd57bf5990efdaf3b941939378d2beacab1df82f867bf5a7e0687badb93"},"schema_version":"1.0","source":{"id":"1507.06871","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.06871","created_at":"2026-05-18T01:36:21Z"},{"alias_kind":"arxiv_version","alias_value":"1507.06871v1","created_at":"2026-05-18T01:36:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06871","created_at":"2026-05-18T01:36:21Z"},{"alias_kind":"pith_short_12","alias_value":"P2O2OBCROKFR","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"P2O2OBCROKFRHLL2","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"P2O2OBCR","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:c6049f85a2d86abc9730fd6abb9bd9ee9bd5429be88eee742ccea2c88b296ce4","target":"graph","created_at":"2026-05-18T01:36:21Z","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 provide a systematic approach to deal with the following problem. Let $X_1,\\ldots,X_n$ be, possibly dependent, $[0,1]$-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper we consider analogues of Hoeffding's result for sums of dependent random variables for which we have certain information on their dependency structure. We prove a result that yields concentration inequalitie","authors_text":"Christos Pelekis, Jan Ramon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-24T15:01:48Z","title":"Hoeffding's inequality for sums of weakly dependent random variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06871","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:db7c4708d16b858ad06906cd69f50e7ade32f7be0065ee8dd6705fae50b2aac5","target":"record","created_at":"2026-05-18T01:36:21Z","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":"0770973cc5f7af844d66b0a43ebcc6bc5aa9783aa869bf1a3d530f5ba732dda5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-24T15:01:48Z","title_canon_sha256":"1de7bcd57bf5990efdaf3b941939378d2beacab1df82f867bf5a7e0687badb93"},"schema_version":"1.0","source":{"id":"1507.06871","kind":"arxiv","version":1}},"canonical_sha256":"7e9da70451728b13ad7a593a6dc91829eda7b0ce36258c5a414caa00664e8296","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7e9da70451728b13ad7a593a6dc91829eda7b0ce36258c5a414caa00664e8296","first_computed_at":"2026-05-18T01:36:21.813834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:21.813834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gndL31+xKKNL7xGDwt9Xc8M5fKzuqCmEJBgwOp9ev4SpLq7cAaPlk9pCV6hUuTmiwoHxb2IXixZxOgfKu1CAAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:21.814263Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.06871","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db7c4708d16b858ad06906cd69f50e7ade32f7be0065ee8dd6705fae50b2aac5","sha256:c6049f85a2d86abc9730fd6abb9bd9ee9bd5429be88eee742ccea2c88b296ce4"],"state_sha256":"1d506c82e91c29ca6794a43aedb23d40b47860ddc6d75214f3182ba9de8c9f3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ygv+iHH1m9aVLmN7Qu3+ioGL8OsYh+xBbXTGQnwZvLzMRmfPZyutuA7jYc+BwjMCIHIrzYcnkz16eD8QEx2OBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T00:52:46.289493Z","bundle_sha256":"d3cf20ab3aa1344bf404d48bee6fcbcc4eb7ed0ddc0cae23740b55d619d4b9c7"}}