Proof of Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Let $\iota$ be the canonical map from $\mathbb{N}$ to $\mathbb{R}$, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. For a Borel map $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ write

Borel and nonnegative by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, in agreement with the notation $D_{T}$ of the statement. For every $\pi'\in\Pi(\mu,\nu)$ the change-of-variables formula, with $(\mathrm{pr}_{1})_{\#}\pi'=\mu$ and $(\mathrm{pr}_{2})_{\#}\pi'=\nu$, gives

Step 1 (Comparison of two maps). Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel with $\int\lVert S\rVert^{2}\,d\mu<\infty$ and let $\pi'\in\Pi(\mu,\nu)$. Writing $a=T(\mathrm{pr}_{1}(z))$, $b=S(\mathrm{pr}_{1}(z))$ and $c=\mathrm{pr}_{2}(z)$, the inequality $\lVert x-y\rVert^{2}\le2\lVert x\rVert^{2}+2\lVert y\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to $x=a-b$ and $y=c-b$, gives $D_{T}(z)\le2\lVert a-b\rVert^{2}+2D_{S}(z)$ pointwise. Integrating against $\pi'$ and using claim 1 of Linearity and Monotonicity of the Lebesgue Integral and $(*)$,

Step 2 (A uniform tail bound). For $j\in\mathbb{N}$ put $R_{j}=\iota(j)$ and $h_{j}(y)=\lVert y\rVert^{2}$ if $R_{j}<\lVert y\rVert^{2}$ and $h_{j}(y)=0$ otherwise; each $h_{j}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, satisfies $0\le h_{j}\le\lVert\cdot\rVert^{2}$, and converges pointwise to $0$, since for fixed $y$ claim 1 of The Archimedean Property of the Real Numbers gives $j$ with $\lVert y\rVert^{2}<\iota(j)$ and claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field makes $h_{j'}(y)=0$ for every $j'\ge j$. As $\lVert\cdot\rVert^{2}$ is integrable with respect to $\nu$, claim 7 of The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere gives that $\bigl(\int_{\mathbb{R}^{d}}h_{j}\,d\nu\bigr)_{j\in\mathbb{N}}$ converges to $0$.

Step 3 (The limit of a weakly convergent subsequence is the coupling induced by $T$). Let $(\sigma_{k})_{k\in\mathbb{N}}$ be a sequence in $\Pi(\mu,\nu)$ with $(I(\sigma_{k}))_{k}$ converging to $W_{2}(\mu,\nu)^{2}$. By Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment §couplings the set $\Pi(\mu,\nu)$ is a tight family, so Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence provides a subsequence $(\sigma_{k_{j}})_{j}$ converging weakly to some $\pi\in\mathcal{P}(\mathbb{R}^{d+d})$, and $\pi\in\Pi(\mu,\nu)$ by The Couplings of Two Probability Measures on Euclidean Space are Closed under Weak Convergence.

Let $\eta$ be a positive real number. Since $(I(\sigma_{k_{j}}))_{j}$ converges to $W_{2}(\mu,\nu)^{2}$ by A Subsequence of a Convergent Sequence Has the Same Limit, there is $j_{0}$ with $I(\sigma_{k_{j}})\le W_{2}(\mu,\nu)^{2}+\eta$ for every $j\ge j_{0}$. The sequence $(\sigma_{k_{j}})_{j\ge j_{0}}$ still converges weakly to $\pi$, so Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit §limit gives $I(\pi)\le W_{2}(\mu,\nu)^{2}+\eta$. As $\eta$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives $I(\pi)\le W_{2}(\mu,\nu)^{2}$; the reverse inequality holds because $W_{2}(\mu,\nu)^{2}$ is the greatest lower bound of the costs of couplings of $\mu$ and $\nu$, by The Quadratic Wasserstein Distance on Euclidean Space §distance. Hence $I(\pi)=W_{2}(\mu,\nu)^{2}$ and $\pi$ is optimal, so $\pi=(\mathrm{id},T)_{\#}\mu$ by unique mapping.

Step 4 (The subsequence has vanishing discrepancy). Keep the notation of Step 3 and let $\eta$ be a positive real number. By The Bounded Lipschitz Vector Fields are Dense in the Square-Integrable Vector Fields Against a Probability Measure §dense there is a bounded Lipschitz $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$, say with $\lVert S\rVert\le M_{S}$, such that $\int_{\mathbb{R}^{d}}\lVert T-S\rVert^{2}\,d\mu\le\eta$.

The function $D_{S}$ is continuous: $S$ is continuous by A Lipschitz Map is Uniformly Continuous, the projections are continuous by claim 1 of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit, and $D_{S}$ is obtained from them by composition and by the continuous map $v\mapsto\lVert v\rVert^{2}$, continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map gives the composition. Moreover

pointwise, by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions.

By Step 2 choose $j_{1}\in\mathbb{N}$ with $\int_{\mathbb{R}^{d}}h_{j_{1}}\,d\nu\le\eta$ and with $2M_{S}^{2}\le R_{j_{1}}$; the order of choice is $\eta$, then $S$ and $M_{S}$, then $j_{1}$. Put $R=R_{j_{1}}$ and $K=4R$, and let $g=\min(D_{S},K)$, which is continuous by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space and satisfies $0\le g\le K$, hence is bounded.

Let $\pi'\in\Pi(\mu,\nu)$ and let $E=\{z:K<D_{S}(z)\}$, a Borel set. For $z\in E$ one has $4R<D_{S}(z)\le2M_{S}^{2}+2\lVert\mathrm{pr}_{2}(z)\rVert^{2}\le R+2\lVert\mathrm{pr}_{2}(z)\rVert^{2}$, so $R<\lVert\mathrm{pr}_{2}(z)\rVert^{2}$ and therefore $2M_{S}^{2}\le R\le\lVert\mathrm{pr}_{2}(z)\rVert^{2}$, whence $D_{S}(z)\le3\lVert\mathrm{pr}_{2}(z)\rVert^{2}=3h_{j_{1}}(\mathrm{pr}_{2}(z))$. Since $D_{S}=g$ off $E$ and $D_{S}\le3\,h_{j_{1}}\circ\mathrm{pr}_{2}$ on $E$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, together with the change-of-variables formula and $(\mathrm{pr}_{2})_{\#}\pi'=\nu$, gives

Using $\pi=(\mathrm{id},T)_{\#}\mu$ together with $g\le D_{S}$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral and the change-of-variables formula,

Since $g$ is bounded and continuous and $(\sigma_{k_{j}})_{j}$ converges weakly to $\pi$, the sequence $\bigl(\int g\,d\sigma_{k_{j}}\bigr)_{j}$ converges to $\int g\,d\pi\le\eta$, so there is $j_{2}$ with $\int g\,d\sigma_{k_{j}}\le2\eta$ for every $j\ge j_{2}$. Combining this with $(\dagger)$ applied to $\sigma_{k_{j}}$ and then with $(**)$,

Step 5 (Claim 1). Suppose the sequence $\bigl(\int D_{T}\,d\pi_{n}\bigr)_{n\in\mathbb{N}}$ does not converge to $0$. Its terms are nonnegative, so there are a positive real $\varepsilon$ and a subsequence $(\pi_{n_{k}})_{k\in\mathbb{N}}$ with $\varepsilon\le\int D_{T}\,d\pi_{n_{k}}$ for every $k$. By A Subsequence of a Convergent Sequence Has the Same Limit the costs $I(\pi_{n_{k}})$ converge to $W_{2}(\mu,\nu)^{2}$, so Steps 3 and 4 apply to $\sigma_{k}=\pi_{n_{k}}$ and yield, for every positive real $\eta$, a further subsequence along which $\int D_{T}\,d\sigma_{k_{j}}\le12\eta$ eventually. Taking $\eta=\tfrac{\varepsilon}{24}$ produces an index $j$ with $\int D_{T}\,d\pi_{n_{k_{j}}}\le\tfrac{\varepsilon}{2}<\varepsilon$, contradicting the choice of the subsequence. Hence $\bigl(\int D_{T}\,d\pi_{n}\bigr)_{n}$ converges to $0$, which is claim 1.