{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:ETVKFJEPQRPCBF4O32HIFWQISC","short_pith_number":"pith:ETVKFJEP","schema_version":"1.0","canonical_sha256":"24eaa2a48f845e20978ede8e82da0890807e7d29e5f5123b07fc7b7190c5b872","source":{"kind":"arxiv","id":"math/0410155","version":1},"attestation_state":"computed","paper":{"title":"Algebraic methods toward higher-order probability inequalities, II","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Donald St. P. Richards","submitted_at":"2004-10-06T12:19:26Z","abstract_excerpt":"Let (L,\\preccurlyeq) be a finite distributive lattice, and suppose that the functions f_1,f_2:L\\to R are monotone increasing with respect to the partial order \\preccurlyeq. Given \\mu a probability measure on L, denote by E(f_i) the average of f_i over L with respect to \\mu, i=1,2. Then the\n FKG inequality provides a condition on the measure \\mu under which the covariance, Cov(f_1,f_2):=E(f_1f_2)-E(f_1)E(f_2), is nonnegative. In this paper we derive a ``third-order'' generalization of the FKG inequality.\n We also establish fourth- and fifth-order generalizations of the FKG inequality and formul"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0410155","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T12:19:26Z","cross_cats_sorted":[],"title_canon_sha256":"3b2bf4d723542c2bcbacab8b05fe1828f9f32eb1b0045489309ea418c33b3ec1","abstract_canon_sha256":"678fcf71f5cfe63740e2791113a89392a04041723c104c0c0e99dd391495b73a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:25.829480Z","signature_b64":"bjsxYB/57xhv9nLnSERHVoczpHdwNNqhgZEwDcwre2jSbN0hfMRPRR2mGy3HGUynrZvTifgLqExwC3HWRaPiBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24eaa2a48f845e20978ede8e82da0890807e7d29e5f5123b07fc7b7190c5b872","last_reissued_at":"2026-05-18T01:05:25.828739Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:25.828739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic methods toward higher-order probability inequalities, II","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Donald St. P. Richards","submitted_at":"2004-10-06T12:19:26Z","abstract_excerpt":"Let (L,\\preccurlyeq) be a finite distributive lattice, and suppose that the functions f_1,f_2:L\\to R are monotone increasing with respect to the partial order \\preccurlyeq. Given \\mu a probability measure on L, denote by E(f_i) the average of f_i over L with respect to \\mu, i=1,2. Then the\n FKG inequality provides a condition on the measure \\mu under which the covariance, Cov(f_1,f_2):=E(f_1f_2)-E(f_1)E(f_2), is nonnegative. In this paper we derive a ``third-order'' generalization of the FKG inequality.\n We also establish fourth- and fifth-order generalizations of the FKG inequality and formul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410155","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0410155","created_at":"2026-05-18T01:05:25.828852+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0410155v1","created_at":"2026-05-18T01:05:25.828852+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410155","created_at":"2026-05-18T01:05:25.828852+00:00"},{"alias_kind":"pith_short_12","alias_value":"ETVKFJEPQRPC","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"ETVKFJEPQRPCBF4O","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"ETVKFJEP","created_at":"2026-05-18T12:25:52.687210+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC","json":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC.json","graph_json":"https://pith.science/api/pith-number/ETVKFJEPQRPCBF4O32HIFWQISC/graph.json","events_json":"https://pith.science/api/pith-number/ETVKFJEPQRPCBF4O32HIFWQISC/events.json","paper":"https://pith.science/paper/ETVKFJEP"},"agent_actions":{"view_html":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC","download_json":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC.json","view_paper":"https://pith.science/paper/ETVKFJEP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0410155&json=true","fetch_graph":"https://pith.science/api/pith-number/ETVKFJEPQRPCBF4O32HIFWQISC/graph.json","fetch_events":"https://pith.science/api/pith-number/ETVKFJEPQRPCBF4O32HIFWQISC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC/action/storage_attestation","attest_author":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC/action/author_attestation","sign_citation":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC/action/citation_signature","submit_replication":"https://pith.science/pith/ETVKFJEPQRPCBF4O32HIFWQISC/action/replication_record"}},"created_at":"2026-05-18T01:05:25.828852+00:00","updated_at":"2026-05-18T01:05:25.828852+00:00"}