{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:PT4VJ4NYNBXB35JK7NAKLLIOII","short_pith_number":"pith:PT4VJ4NY","schema_version":"1.0","canonical_sha256":"7cf954f1b8686e1df52afb40a5ad0e4239e47ccaca25c70c2803cb14d36efb0f","source":{"kind":"arxiv","id":"math/0607686","version":2},"attestation_state":"computed","paper":{"title":"The Modulo 1 Central Limit Theorem and Benford's Law for Products","license":"","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Mark J. Nigrini, Steven J. Miller","submitted_at":"2006-07-26T20:18:50Z","abstract_excerpt":"We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence is that if X_1, ..., X_M are independent continuous random variables with densities f_1, ..., f_M, for any base B as M \\to \\infty for many choices of the densities the distribution of the digits of X_1 * ... * X_M converges to Benford's law base B. The rate of convergence can be quantified in terms of the Fourier coefficients of the densities, and provides a"},"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/0607686","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2006-07-26T20:18:50Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"90b71fcd747f23f6cc6dedc40fc3e9e2fcd8867482348875dd603d1ca955bd51","abstract_canon_sha256":"ba88ef26b18353f266e847509880c6d7e28c22b0383a5cb6f34d6530b9fce9b7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:56.379701Z","signature_b64":"hGwh8XrUo5i+cRIIHhLOhjyp2ANOWBSj7KYHsB0ceI1+wmaCQnfC+sNgvfs/WlTIwXSBIR1mtTXzunc996GOAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cf954f1b8686e1df52afb40a5ad0e4239e47ccaca25c70c2803cb14d36efb0f","last_reissued_at":"2026-05-18T04:40:56.379122Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:56.379122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Modulo 1 Central Limit Theorem and Benford's Law for Products","license":"","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Mark J. Nigrini, Steven J. Miller","submitted_at":"2006-07-26T20:18:50Z","abstract_excerpt":"We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence is that if X_1, ..., X_M are independent continuous random variables with densities f_1, ..., f_M, for any base B as M \\to \\infty for many choices of the densities the distribution of the digits of X_1 * ... * X_M converges to Benford's law base B. The rate of convergence can be quantified in terms of the Fourier coefficients of the densities, and provides a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607686","kind":"arxiv","version":2},"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/0607686","created_at":"2026-05-18T04:40:56.379224+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0607686v2","created_at":"2026-05-18T04:40:56.379224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607686","created_at":"2026-05-18T04:40:56.379224+00:00"},{"alias_kind":"pith_short_12","alias_value":"PT4VJ4NYNBXB","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"PT4VJ4NYNBXB35JK","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"PT4VJ4NY","created_at":"2026-05-18T12:25:54.717736+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/PT4VJ4NYNBXB35JK7NAKLLIOII","json":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII.json","graph_json":"https://pith.science/api/pith-number/PT4VJ4NYNBXB35JK7NAKLLIOII/graph.json","events_json":"https://pith.science/api/pith-number/PT4VJ4NYNBXB35JK7NAKLLIOII/events.json","paper":"https://pith.science/paper/PT4VJ4NY"},"agent_actions":{"view_html":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII","download_json":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII.json","view_paper":"https://pith.science/paper/PT4VJ4NY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0607686&json=true","fetch_graph":"https://pith.science/api/pith-number/PT4VJ4NYNBXB35JK7NAKLLIOII/graph.json","fetch_events":"https://pith.science/api/pith-number/PT4VJ4NYNBXB35JK7NAKLLIOII/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII/action/storage_attestation","attest_author":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII/action/author_attestation","sign_citation":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII/action/citation_signature","submit_replication":"https://pith.science/pith/PT4VJ4NYNBXB35JK7NAKLLIOII/action/replication_record"}},"created_at":"2026-05-18T04:40:56.379224+00:00","updated_at":"2026-05-18T04:40:56.379224+00:00"}