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

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