{"paper":{"title":"A Simple Direct Proof of Billingsley's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fred Kochman, Richard Arratia","submitted_at":"2014-01-08T00:57:22Z","abstract_excerpt":"Billingsley's theorem (1972) asserts that the Poisson--Dirichlet process is the limit, as $n \\to \\infty$, of the process giving the relative log sizes of the largest prime factor, the second largest, and so on, of a random integer chosen uniformly from 1 to $n$. In this paper we give a new proof that directly exploits Dickman's asymptotic formula for the number of such integers with no prime factor larger than $n^{1/u}$, namely $\\Psi(n,n^{1/u}) \\sim n \\rho(u)$, to derive the limiting joint density functions of the finite-dimensional projections of the log prime factor processes. Our main techn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1553","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"}