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Billingsley's Theorem (1972), in its modern formulation, asserts that the limiting process, as $n \\to \\infty$, is the Poisson-Dirichlet process with parameter $\\theta =1$.\n  In this paper we give a new proof, inspired by the 1993 proof by Donnelly and Grimmett, and extend the result to factorizations of elements of normed arithmetic semigroups satisfying certain growth conditions, for "},"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":"1401.1555","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-08T01:03:26Z","cross_cats_sorted":[],"title_canon_sha256":"0faa9a81b52da1b6623c8efe2fec6996f056e436f5cd52a32784b2cec3253259","abstract_canon_sha256":"a063b2398888c77e36e9706eaf740b5e11e5e6db65f7194eee70e14fdcad6878"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:58.638447Z","signature_b64":"vpH5D2EQLp0Y6iNCtUPurBtDfRRUEvfCUxibJ1bLO9G4azAWtaYqROAsiW4qrpE5FdEHZDHxsVh3l3IOFK4ECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ccd65ba152077b39adc06eaf42591728c58b01a38b2806abbaca3d510b31dba","last_reissued_at":"2026-05-18T03:02:58.637745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:58.637745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extensions of Billingsley's Theorem via Multi-Intensities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fred Kochman, Richard Arratia, Victor S. 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Billingsley's Theorem (1972), in its modern formulation, asserts that the limiting process, as $n \\to \\infty$, is the Poisson-Dirichlet process with parameter $\\theta =1$.\n  In this paper we give a new proof, inspired by the 1993 proof by Donnelly and Grimmett, and extend the result to factorizations of elements of normed arithmetic semigroups satisfying certain growth conditions, for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1555","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":"1401.1555","created_at":"2026-05-18T03:02:58.637858+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.1555v1","created_at":"2026-05-18T03:02:58.637858+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.1555","created_at":"2026-05-18T03:02:58.637858+00:00"},{"alias_kind":"pith_short_12","alias_value":"LTGWLOQVEB33","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"LTGWLOQVEB33HGW4","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"LTGWLOQV","created_at":"2026-05-18T12:28:38.356838+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2402.11884","citing_title":"Large prime factors of well-distributed sequences","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK","json":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK.json","graph_json":"https://pith.science/api/pith-number/LTGWLOQVEB33HGW4A3VPIJMROK/graph.json","events_json":"https://pith.science/api/pith-number/LTGWLOQVEB33HGW4A3VPIJMROK/events.json","paper":"https://pith.science/paper/LTGWLOQV"},"agent_actions":{"view_html":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK","download_json":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK.json","view_paper":"https://pith.science/paper/LTGWLOQV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.1555&json=true","fetch_graph":"https://pith.science/api/pith-number/LTGWLOQVEB33HGW4A3VPIJMROK/graph.json","fetch_events":"https://pith.science/api/pith-number/LTGWLOQVEB33HGW4A3VPIJMROK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK/action/storage_attestation","attest_author":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK/action/author_attestation","sign_citation":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK/action/citation_signature","submit_replication":"https://pith.science/pith/LTGWLOQVEB33HGW4A3VPIJMROK/action/replication_record"}},"created_at":"2026-05-18T03:02:58.637858+00:00","updated_at":"2026-05-18T03:02:58.637858+00:00"}