{"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. Miller","submitted_at":"2014-01-08T01:03:26Z","abstract_excerpt":"Let $p_1 \\ge p_2 \\ge \\dots$ be the prime factors of a random integer chosen uniformly from $1$ to $n$, and let $$ \\frac{\\log p_1}{\\log n}, \\frac{\\log p_2}{\\log n}, \\dots $$ be the sequence of scaled log factors. 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"}