{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:F4AWU5YV4WUGQZKNZ4VSHPRIAB","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":"dd73abeac6790c62b8cff9a31441df8fd84c468dcd7c5c4a9aef84fb9bdbf2be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-09T01:06:59Z","title_canon_sha256":"953d448b92dc708417b3fb880f86d43cd94b6f82f8914e4749247098574a28f4"},"schema_version":"1.0","source":{"id":"1606.02794","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02794","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02794v2","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02794","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"pith_short_12","alias_value":"F4AWU5YV4WUG","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"F4AWU5YV4WUGQZKN","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"F4AWU5YV","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:8aac05e29614d3b003082b561d4f7ff6ae811adb277a5289860dccbc44c5f784","target":"graph","created_at":"2026-05-18T00:38:36Z","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":"Let $\\{X_n\\}_{n\\geq 1}$ be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centered sequence with a suitable level of negative dependence. We prove Baum-Katz type theorems by only assuming that the variables $X_n$ satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centered, independent random variables. This leads to Marcinkiewicz-Zygmund type strong laws of large numbers with estimate for the rate of convergence.","authors_text":"Rich\\'ard Balka, Tibor T\\'om\\'acs","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-09T01:06:59Z","title":"Baum-Katz type theorems with exact threshold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02794","kind":"arxiv","version":2},"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:e18454ae1f2e4e2b06fce871e3b479ddb542111a788b21275bb5cf5bba07a2e3","target":"record","created_at":"2026-05-18T00:38:36Z","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":"dd73abeac6790c62b8cff9a31441df8fd84c468dcd7c5c4a9aef84fb9bdbf2be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-09T01:06:59Z","title_canon_sha256":"953d448b92dc708417b3fb880f86d43cd94b6f82f8914e4749247098574a28f4"},"schema_version":"1.0","source":{"id":"1606.02794","kind":"arxiv","version":2}},"canonical_sha256":"2f016a7715e5a868654dcf2b23be280057eb3f1354fbcd57af92590da5790afe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f016a7715e5a868654dcf2b23be280057eb3f1354fbcd57af92590da5790afe","first_computed_at":"2026-05-18T00:38:36.476204Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:36.476204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z3hYeP7bqtyq7e+JVyXxBGnvxlBqXz2UeDzBOrQ4V40wk13hlokd9emTF+McA1P+tpHvLzP+5UbpImSw+OxXBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:36.476698Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02794","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e18454ae1f2e4e2b06fce871e3b479ddb542111a788b21275bb5cf5bba07a2e3","sha256:8aac05e29614d3b003082b561d4f7ff6ae811adb277a5289860dccbc44c5f784"],"state_sha256":"dd7fd54bf181702adb7c709552496683e232750db07f0f7f8f1bc045d8e3aa94"}