The topological support of the z-measures on the Thoma simplex

TL;DR

This paper proves that the support of nondegenerate z-measures on the Thoma simplex is the entire space, linking these measures to Poisson-Dirichlet distributions and using advanced topological and measure-theoretic techniques.

Contribution

It establishes that the support of any nondegenerate z-measure on the Thoma simplex is the whole space, clarifying their topological properties.

Findings

Support of nondegenerate z-measures is the entire Thoma simplex

Connection between z-measures and Poisson-Dirichlet distributions

Use of advanced topological and measure-theoretic methods

Abstract

The Thoma simplex $\Omega$ is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are a family of probability measures on $\Omega$ depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit when it goes to $0$, the z-measures turn into the Poisson-Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space $\Omega$. The proof is based on results of arXiv:0902.3395 and arXiv:1806.07454.