On a completeness problem in a Fourier-based probability metrics in $\mathbb{R}^N$

TL;DR

This paper investigates the completeness of probability measure spaces with specific Fourier-based metrics, proving completeness for even moments and providing counterexamples for odd moments, thus resolving an open problem from 2007.

Contribution

It establishes the completeness of certain probability measure spaces with Fourier-based metrics for even moments and constructs counterexamples for odd moments, solving a longstanding open problem.

Findings

Spaces are complete for even s

Counterexamples show incompleteness for odd s

Solves an open problem from 2007

Abstract

We study completeness of the spaces $\mathcal{P}_s^=$ of probability measures in $\mathbb{R}^N$ which have equal (prescribed) moments up to order $s \in \mathbb{N}$, endowed with the metric $d_s(\mu,\nu)=\sup_{x \in \mathbb{R}^N\setminus 0}\frac{|\hat \mu(x)-\hat \nu(x)|}{|x|^s}$, where $\hat \mu$ is the characteristic function of $\mu$. We prove that the spaces $(\mathcal{P}_s^=,d_s)$ are complete if $s$ is even and construct suitable counterexamples to completeness for all odd $s$. This solves an open problem formulated by J. Carrillo and G. Toscani in 2007.