Universality of Asymptotically Ewens Measures on Partitions

TL;DR

This paper establishes a universality theorem for measures on partitions that resemble the Ewens measure, extending key limit theorems and simplifying proofs through a unified criterion and the Feller coupling.

Contribution

It extends the class of measures for which classical partition limit theorems hold, providing a simple criterion for universality and streamlining the proofs using the Feller coupling.

Findings

Limit theorems for longest parts extend to asymptotically Ewens measures

Functional central limit theorem for the number of parts applies broadly

Erdos-Turan limit for the product of parts is generalized

Abstract

We introduce a universality theorem for functionals of measures on partitions which "behave like" the Ewens measure. Various limit theorems for the Ewens measure, most notably the Poisson-Dirichlet limit for the longest parts, the functional central limit theorem for the number of parts, and the Erdos-Turan limit for the product of parts, extend to these asymptotically Ewens measures as easy corollaries. Our major contributions are: (1) extending the classes of measures for which these limit theorems hold; (2) characterising universality by a single, easily-checked criterion; and (3) greatly shortening the proofs of the limit theorems using the Feller coupling.