{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:CG7LF3XFW4RYOXIZWLEAPNWZWK","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":"bded4e86132dfaeef7662c041c42f77828a2463c87fadf5dc2c89fd25a112d2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-04-23T17:18:34Z","title_canon_sha256":"369f2336b7af548a9b55d4d8b42766068cc84eb4b00f6251447f87d85b63d4a2"},"schema_version":"1.0","source":{"id":"2004.11324","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2004.11324","created_at":"2026-07-05T03:04:16Z"},{"alias_kind":"arxiv_version","alias_value":"2004.11324v3","created_at":"2026-07-05T03:04:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2004.11324","created_at":"2026-07-05T03:04:16Z"},{"alias_kind":"pith_short_12","alias_value":"CG7LF3XFW4RY","created_at":"2026-07-05T03:04:16Z"},{"alias_kind":"pith_short_16","alias_value":"CG7LF3XFW4RYOXIZ","created_at":"2026-07-05T03:04:16Z"},{"alias_kind":"pith_short_8","alias_value":"CG7LF3XF","created_at":"2026-07-05T03:04:16Z"}],"graph_snapshots":[{"event_id":"sha256:5a90ff058cae3edc7c116de1d297c9c0d82a772349f8adcc3aae6bc4002ec438","target":"graph","created_at":"2026-07-05T03:04:16Z","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/2004.11324/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The converse of the Borel-Cantelli Lemma states that if $\\{A_i\\}_{i=1}^\\infty$ is a sequence of independent events such that $\\sum P(A_i)=\\infty$, then almost surely infinitely many of these events will occur. Erd\\H os and R\\'enyi proved that it is sufficient to weaken the condition of independence to pairwise independence. In this paper we study various conditions of weak independence that imply the conclusion of the converse of the Borel--Cantelli Lemma. We will determine the exact implicational relationship among these conditions.","authors_text":"Csaba Bir\\'o, Israel R. Curbelo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-04-23T17:18:34Z","title":"Weak independence of events and the converse of the Borel--Cantelli Lemma"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.11324","kind":"arxiv","version":3},"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:bf81c8f21b434b93fa765a1e3872492194881064003c8f1c57ff418d3e8fe38b","target":"record","created_at":"2026-07-05T03:04:16Z","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":"bded4e86132dfaeef7662c041c42f77828a2463c87fadf5dc2c89fd25a112d2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-04-23T17:18:34Z","title_canon_sha256":"369f2336b7af548a9b55d4d8b42766068cc84eb4b00f6251447f87d85b63d4a2"},"schema_version":"1.0","source":{"id":"2004.11324","kind":"arxiv","version":3}},"canonical_sha256":"11beb2eee5b723875d19b2c807b6d9b28f8032971f48e4ee392a3f5419abd51c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11beb2eee5b723875d19b2c807b6d9b28f8032971f48e4ee392a3f5419abd51c","first_computed_at":"2026-07-05T03:04:16.071245Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:04:16.071245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R0djGLFUKzebwEAiwETQEUmxjA2i2yX+JWZWMrXR/c8jL/95Jorp/2uHFKr+DHSTNUANIHsWJLTd1Y/2MC/VDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:04:16.071629Z","signed_message":"canonical_sha256_bytes"},"source_id":"2004.11324","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf81c8f21b434b93fa765a1e3872492194881064003c8f1c57ff418d3e8fe38b","sha256:5a90ff058cae3edc7c116de1d297c9c0d82a772349f8adcc3aae6bc4002ec438"],"state_sha256":"7c289b481465500eac5b7c4ad35f93b04810f720c9ed7b22ddad1f92cbe049a8"}