Can I read A Simple Direct Proof of Billingsley's Theorem on EtoBox?

A Simple Direct Proof of Billingsley's Theorem by Arratia, Richard; Kochman, Fred is a scholarly article available to read on EtoBox.

What is A Simple Direct Proof of Billingsley's Theorem about?

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.

Author

Arratia, Richard; Kochman, Fred

Published

2014

Language

EN