The full moment problem on subsets of probabilities and point configurations

TL;DR

This paper investigates the full moment problem for measures supported on specific non-linear subsets of an infinite-dimensional space, focusing on random measures like point configurations, and improves existing representation techniques for these sets.

Contribution

It provides new representations of subsets such as point configurations as semi-algebraic sets, enhancing the application of the full moment problem results.

Findings

Representation of point configurations as semi-algebraic sets

Improved conditions for the full moment problem on these sets

Simplified handling of correlation functions for point configurations

Abstract

The aim of this paper is to study the full $K-$moment problem for measures supported on some particular non-linear subsets $K$ of an infinite dimensional vector space. We focus on the case of random measures, that is $K$ is a subset of all non-negative Radon measures on $\mathbb{R}^d$. We consider as $K$ the space of sub-probabilities, probabilities and point configurations on $\mathbb{R}^d$. For each of these spaces we provide at least one representation as a generalized basic closed semi-algebraic set to apply the main result in [J. Funct. Anal., 267 (2014) no.5: 1382--1418]. We demonstrate that this main result can be significantly improved by further considerations based on the particular chosen representation of $K$. In the case when $K$ is a space of point configurations, the correlation functions (also known as factorial moment functions) are easier to handle than the ordinary moment functions. Hence, we additionally express the main results in terms of correlation functions.