Poisson--Dirichlet Limit Theorems in Combinatorial Applications via Multi-Intensities

Abstract

We present new, exceptionally efficient proofs of Poisson--Dirichlet limit theorems for the scaled sizes of irreducible components of random elements in the classic combinatorial contexts of arbitrary assemblies, multisets, and selections, when the components generating functions satisfy certain standard hypotheses. The proofs exploit a new criterion for Poisson--Dirichlet limits, originally designed for rapid proofs of Billingsley's theorem on the scaled sizes of log prime factors of random integers (and some new generalizations). Unexpectedly, the technique applies in the present combinatorial setting as well, giving, perhaps, a long sought-after unifying point of view. The proofs depend also on formulas of Arratia and Tavar{\'e} for the mixed moments of counts of components of various sizes, as well as formulas of Flajolet and Soria for the asymptotics of generating function coefficients.