{"paper":{"title":"Finite-sample Borel--Cantelli inequalities under mixing conditions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Explicit finite-N lower bounds for the union probability of events hold under quantitative mixing conditions.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chatchawan Panraksa","submitted_at":"2026-04-26T16:31:01Z","abstract_excerpt":"We prove explicit one-lag finite-$N$ lower bounds for $\\mathbb P(\\bigcup_{k=1}^N A_k)$ using only the marginal probabilities and a selected-lag dependence coefficient of the event-generated $\\sigma$-fields. Each finite statement uses one fixed coefficient convention, either ambient or finite restricted, rather than a limiting mixing assumption. In the $\\varphi$ case, a residue-class blocking argument and a one-sided approximate-independence inequality yield a free spacing parameter $L\\ge0$, spacing coefficient $1/(L+1)$, and residual terms governed by $\\varphi(L+1)$. In the $\\alpha$ case, we g"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove explicit finite-N lower bounds for P(∪_{k=1}^N A_k) when the σ-algebras generated by an event sequence satisfy quantitative ϕ- or α-mixing bounds. The main ϕ-mixing estimate is obtained by a residue-class blocking argument and a one-sided approximate-independence inequality; it has a free spacing parameter L≥0, spacing coefficient 1/(L+1), and residual terms governed by ϕ(L+1).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The event sequence satisfies quantitative ϕ-mixing or α-mixing bounds (i.e., the mixing coefficients decay at a known rate), which is invoked to control the residual terms after blocking.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit finite-N lower bounds for union probabilities under phi- or alpha-mixing are proved via residue-class blocking with spacing coefficient 1/(L+1) and mixing residuals.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Explicit finite-N lower bounds for the union probability of events hold under quantitative mixing conditions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a2ea6c891bc07375adb5758ccf3f5962fced57faf32211d1c1a9666fc3c58f44"},"source":{"id":"2604.23791","kind":"arxiv","version":2},"verdict":{"id":"736a4d75-11f8-4369-a765-a4683a8dbe2a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T05:29:39.331953Z","strongest_claim":"We prove explicit finite-N lower bounds for P(∪_{k=1}^N A_k) when the σ-algebras generated by an event sequence satisfy quantitative ϕ- or α-mixing bounds. The main ϕ-mixing estimate is obtained by a residue-class blocking argument and a one-sided approximate-independence inequality; it has a free spacing parameter L≥0, spacing coefficient 1/(L+1), and residual terms governed by ϕ(L+1).","one_line_summary":"Explicit finite-N lower bounds for union probabilities under phi- or alpha-mixing are proved via residue-class blocking with spacing coefficient 1/(L+1) and mixing residuals.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The event sequence satisfies quantitative ϕ-mixing or α-mixing bounds (i.e., the mixing coefficients decay at a known rate), which is invoked to control the residual terms after blocking.","pith_extraction_headline":"Explicit finite-N lower bounds for the union probability of events hold under quantitative mixing conditions."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2604.23791/integrity.json","findings":[{"note":"Identifier '10.1016/0304-4149(84' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. The cited work could not be located through any authoritative source.","detector":"doi_compliance","severity":"critical","ref_index":13,"audited_at":"2026-05-19T22:46:26.706115Z","detected_doi":"10.1016/0304-4149(84","finding_type":"unresolvable_identifier","verdict_class":"cross_source","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T08:34:44.619266Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:46:26.706115Z","status":"completed","version":"1.0.0","findings_count":1}],"snapshot_sha256":"fd06c08df7ef24a02df2239cc2dcf2a0aa9016e84d8c42343a23d4c44b0d17c3"},"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"}