A Simple Direct Proof of Billingsley's Theorem

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.

13 pages

References in corpus (1)

• On the Amount of Dependence in the Prime Factorization of a Uniform Random Integer

Cited by in corpus (1)

• Extensions of Billingsley's Theorem via Multi-Intensities