A Simple Direct Proof of Billingsley's Theorem

TL;DR

This paper provides a new, direct proof of Billingsley's theorem on the convergence of the relative sizes of prime factors of random integers, using Dickman's asymptotic formula and a novel convergence criterion.

Contribution

It introduces a new proof method that directly leverages Dickman's formula and a new criterion for convergence in distribution of discrete to continuous variables.

Findings

Established the limiting joint density functions of prime factor size processes.

Provided a new proof technique for convergence in distribution involving non-lattice discrete variables.

Confirmed Billingsley's theorem using the new approach.

Abstract

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 technical tool is a new criterion for the convergence in distribution of non-lattice discrete random variables to continuous random variables.