Independent Elliptical Distributions Minimize Their $\mathcal{W}_2$ Wasserstein Distance from Independent Elliptical Distributions with the Same Density Generator

This short note is on a property of the $\mathcal{W}_2$ Wasserstein distance which indicates that independent elliptical distributions minimize their $\mathcal{W}_2$ Wasserstein distance from given independent elliptical distributions with the same density generators. Furthermore, we examine the implications of this property in the Gelbrich bound when the distributions are not necessarily elliptical. Meanwhile, we also generalize the results to the cases when the distributions are not independent. The primary purpose of this note is for the referencing of papers that need to make use of this property or its implications.