A Simple Direct Proof of Billingsley's Theorem

2014/01/08 by Richard Arratia, Arratia, Richard, Fred Kochman +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1401.1553

13 pages

arxiv created 2014/01/08 · arxiv updated 2014/01/09

Abstract

Billingsley's theorem (1972) asserts that the Poisson--Dirichlet process is the limit, as n → ∞, 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 Ψ(n,n^1/u) ∼ n ρ(u), to derive the limiting joint density functions of the finite-dimensional projections of the log prime factor processes. Our main technical tool is a new criterion for the convergence in distribution of non-lattice discrete random variables to continuous random variables.

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