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

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.

arXiv commentSubstantially revised and expanded; new title; updated references

References in corpus (1)

• Absolute moments and Fourier-based probability metrics