Random Moment Problems under Constraints

TL;DR

This paper studies the behavior of random moment sequences under fixed moment constraints, revealing universal Gaussian fluctuations and deviations, and highlighting the impact of constraints on the connection to random matrix theory.

Contribution

It provides a detailed probabilistic analysis of moment space sections under constraints, including universal fluctuation results and the effect of fixing moments on random matrix connections.

Findings

Gaussian fluctuations of random moments

Universal behavior of barycenter measures

Breaking the link to random matrices under constraints

Abstract

We investigate moment sequences of probability measures on $E\subset\mathbb{R}$ under constraints of certain moments being fixed. This corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of moment sequences of order $n$. By equipping these sections with the uniform or more general probability distributions, we manage to give for large $n$ precise results on the (probabilistic) barycenters of moment space sections and the fluctuations of random moments around these barycenters. The measures associated to the barycenters belong to the Bernstein-Szeg\H{o} class and show strong universal behavior. We prove Gaussian fluctuations and moderate and large deviations principles. Furthermore, we demonstrate how fixing moments by a constraint leads to breaking the connection between random moments and random matrices.