  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.
