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

Each result cited is universally quantified over the data in its own statement. Elements of $L^{2}(\Omega;\mathbb{R}^{d})$ are written through representatives in $\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors (v); for representatives $X,Y$, the class $X-Y$ is $[X-Y]$ and $\lVert X-Y\rVert_{L^{2}}^{2}=\mathbb{E}[\lVert X-Y\rVert^{2}]$, by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (v) (the additive inverse of $[Y]$ being $[(-1)Y]$), The Space of Square-Integrable Random Vectors (ii), claims 2 and 3 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space (giving $X+(-1)Y=X-Y$) and The Space of Square-Integrable Random Vectors (iii). By The Quadratic Wasserstein Distance on Euclidean Space (i), $W_{2}(\mu,\nu)$ is nonnegative and $W_{2}(\mu,\nu)^{2}\le I(\pi)$ for every $\pi\in\Pi(\mu,\nu)$.

Claim 1. By The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii), $\mathcal{L}(X)\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for every class $X$, and the law is that of the class by The Space of Square-Integrable Random Vectors (iv).

Claim 2. Let $X,Y\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ represent the two classes. By Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v), the law of the pairing $(X,Y)$ is a coupling of $\mathcal{L}(X)$ and $\mathcal{L}(Y)$ with quadratic cost $\mathbb{E}[\lVert X-Y\rVert^{2}]=\lVert X-Y\rVert_{L^{2}}^{2}$. Hence $W_{2}(\mathcal{L}(X),\mathcal{L}(Y))^{2}\le\lVert X-Y\rVert_{L^{2}}^{2}$, and since both $W_{2}(\mathcal{L}(X),\mathcal{L}(Y))$ and $\lVert X-Y\rVert_{L^{2}}$ are nonnegative (the latter by Real Inner Product Spaces (ii)), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\mathcal{L}(X),\mathcal{L}(Y))\le\lVert X-Y\rVert_{L^{2}}$.

Claim 3. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Rich Probability Space (i) with $m=d$ there is a random vector $X$ in $\mathbb{R}^{d}$ on $(\Omega,\mathcal{F},P)$ with $\mathcal{L}(X)=\mu$. By Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii) applied to the Borel map $x\mapsto\lVert x\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), read as $[0,\infty]$-valued, $\mathbb{E}[\lVert X\rVert^{2}]=\int\lVert x\rVert^{2}\,\mu(dx)=M_{2}(\mu)<\infty$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i) and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (ii); so $X\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors (i), and its class has law $\mu$ by The Space of Square-Integrable Random Vectors (iv).

Claim 4. By claim 3 there are classes $X,Y$ with laws $\mu$ and $\nu$, so $\lVert X-Y\rVert_{L^{2}}\in D(\mu,\nu)$ and $D(\mu,\nu)$ is nonempty; every member of $D(\mu,\nu)$ is at least $W_{2}(\mu,\nu)$ by claim 2, so $D(\mu,\nu)$ is bounded below by $W_{2}(\mu,\nu)$ and has a greatest lower bound $\inf D(\mu,\nu)$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, with $W_{2}(\mu,\nu)\le\inf D(\mu,\nu)$ by Lower Bound and Greatest Lower Bound. For the reverse inequality, write $W=W_{2}(\mu,\nu)$ and let $\varepsilon\in\mathbb{R}$ satisfy $0<\varepsilon$. Put $\eta=2W\varepsilon+\varepsilon^{2}$; here $0\le2W\varepsilon$ by claim 5 of Elementary Arithmetic in an Ordered Field ($W$ being nonnegative and $0<2\varepsilon$ by claims 8 and 5 of Elementary Order Arithmetic in an Ordered Field), $0<\varepsilon^{2}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, so $0<\eta$ by claim 3 there; and $\eta$ satisfies $W^{2}+\eta=(W+\varepsilon)^{2}$ by claim 5 of Zero Products and Elementary Identities in a Field. By The Quadratic Wasserstein Distance on Euclidean Space (i), $W^{2}$ is the greatest lower bound of $\{I(\pi):\pi\in\Pi(\mu,\nu)\}$, so claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ provides $\pi\in\Pi(\mu,\nu)$ with $I(\pi)<W^{2}+\eta=(W+\varepsilon)^{2}$, whence $\sqrt{I(\pi)}<W+\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By Rich Probability Space (i) with $m=d+d$ there is a random vector $Z$ in $\mathbb{R}^{d+d}$ with $\mathcal{L}(Z)=\pi$, and by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v) $Z=(X',Y')$ for random vectors $X'=\mathrm{pr}_{1}\circ Z$, $Y'=\mathrm{pr}_{2}\circ Z$ in $\mathbb{R}^{d}$, whose laws satisfy $\mathcal{L}(X')=(\mathrm{pr}_{1})_{\#}\pi=\mu$ and $\mathcal{L}(Y')=(\mathrm{pr}_{2})_{\#}\pi=\nu$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii) and Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i), and $\mathbb{E}[\lVert X'-Y'\rVert^{2}]=I(\pi)$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v). As in claim 3, $X',Y'\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$, and $\lVert X'-Y'\rVert_{L^{2}}=\sqrt{I(\pi)}<W+\varepsilon$ belongs to $D(\mu,\nu)$, the identification of the norm with $\sqrt{I(\pi)}$ following from $\lVert X'-Y'\rVert_{L^{2}}^{2}=\mathbb{E}[\lVert X'-Y'\rVert^{2}]=I(\pi)$ and the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. Hence $\inf D(\mu,\nu)<W+\varepsilon$ for every positive $\varepsilon$; if $W<\inf D(\mu,\nu)$ held, the choice $\varepsilon=\inf D(\mu,\nu)-W$, positive by claim 3 of Elementary Arithmetic in an Ordered Field and claim 1 of Elementary Order Arithmetic in an Ordered Field, would give $\inf D(\mu,\nu)<\inf D(\mu,\nu)$, which is impossible. Therefore $\inf D(\mu,\nu)\le W$, and $W_{2}(\mu,\nu)=\inf D(\mu,\nu)$.