The Wasserstein Distance and the Mean-Square Distance of Random Vectors

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$. Let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors, with its norm $\lVert\cdot\rVert_{L^{2}}$, the law $\mathcal{L}(X)$ of a class and the notational convention fixed there; let $\mathcal{P}_{2}(\mathbb{R}^{d})$ be the set of probability measures with finite second moment and $W_{2}$ the quadratic Wasserstein distance. Then the following hold.

1. (The law map)(i) The map $\Lambda:L^{2}(\Omega;\mathbb{R}^{d})\to\mathcal{P}_{2}(\mathbb{R}^{d})$, $\Lambda(X)=\mathcal{L}(X)$, is well defined, by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii).

2. (The Wasserstein distance is dominated by the mean-square distance)(ii) For all $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$,

$$W_{2}\bigl(\mathcal{L}(X),\mathcal{L}(Y)\bigr)\le\lVert X-Y\rVert_{L^{2}} .$$

3. (Surjectivity on a rich space)(iii) If $(\Omega,\mathcal{F},P)$ is rich, then for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ there is $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$; that is, $\Lambda$ is surjective.

4. (The distance as an infimum over random vectors)(iv) If $(\Omega,\mathcal{F},P)$ is rich, then for all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the set $D(\mu,\nu)=\{\lVert X-Y\rVert_{L^{2}}:X,Y\in L^{2}(\Omega;\mathbb{R}^{d}),\ \mathcal{L}(X)=\mu,\ \mathcal{L}(Y)=\nu\}$ is a nonempty set of real numbers bounded below by $W_{2}(\mu,\nu)$, hence has a greatest lower bound $\inf D(\mu,\nu)$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, and

$$W_{2}(\mu,\nu)=\inf D(\mu,\nu).$$