Proof of Wasserstein Convergence on a Hilbert Space: Weak Convergence, Integrals of Continuous Functions of Quadratic Growth, Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Compactness

Each result cited below is universally quantified over the data in its own statement and is applied with the data named at the point of citation.

Conventions. $\Pi(\rho,\sigma)$ and $I$ are the couplings and the quadratic cost of that definition, and $W_{2}(\rho,\sigma)^{2}\le I(\pi)$ for every $\pi\in\Pi(\rho,\sigma)$ by The Quadratic Wasserstein Distance on a Hilbert Space §distance. $\mathbb{R}$ carries the absolute-value metric $d_{\mathbb{R}}$. A continuous map from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$ is Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space, a Lipschitz map between metric spaces is continuous by A Lipschitz Map is Uniformly Continuous, and by claim 6 of Borel Measurability and Bounded Integration on a Metric Space every bounded Borel function on $X$ is integrable with respect to every member of $\mathcal{P}(X)$, constants integrating to themselves, with the integral of a nonnegative one equal to its integral in $[0,\infty]$. Integrals of nonnegative Borel functions are taken in $[0,\infty]$; for a nonnegative integrable function the two integrals agree, its negative part vanishing in Integrable Function and the Lebesgue Integral, and a Borel $f$ is integrable exactly when $\int|f|<\infty$ by the criterion recorded there. Additivity and monotonicity of integrals are Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and linearity for integrable functions is Linearity and Monotonicity of the Lebesgue Integral §integrable. The function $x\mapsto|x|^{2}$ on $X$ is continuous and Borel as recorded in The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment, and $M_{2}$ is the second moment. Weak convergence is that of Weak Convergence of Finite Borel Measures on a Metric Space, and limits of real sequences are those of Limit of a Sequence of Real Numbers. $X$ is nonempty, as $0_{X}\in X$.

Step 1 (Claim 1). By Existence of an Optimal Coupling of Two Borel Probability Measures with Finite Second Moment on a Hilbert Space §existence, for each $j$ there is $\pi_{j}\in\Pi(\mu_{j},\mu)$ with $I(\pi_{j})=W_{2}(\mu_{j},\mu)^{2}$, a real number. Let $f:X\to\mathbb{R}$ be bounded and Lipschitz from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$ with some constant $L\ge0$, which is the notion of Lipschitz function in the preamble of Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound. By Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §lipschitz, $f$ is Borel and integrable with respect to $\mu_{j}$ and $\mu$, and

$$\Bigl|\int_{X}f\,d\mu_{j}-\int_{X}f\,d\mu\Bigr|\le L\sqrt{I(\pi_{j})}=L\,W_{2}(\mu_{j},\mu).$$

The sequence $(L\,W_{2}(\mu_{j},\mu))_{j}$ converges to $0$ by claim 3 of Arithmetic of Limits of Real Sequences, so $(\int_{X}f\,d\mu_{j})_{j}$ converges to $\int_{X}f\,d\mu$ by claim 3 of Order Properties of Limits of Real Sequences. As $X$ is nonempty and $\mu_{j},\mu$ are probability measures on $(X,\mathcal{B}(X))$, claim 1 of Portmanteau Theorem on a Metric Space gives $\mu_{j}\Rightarrow\mu$.

Step 2 (Claim 2). (i) The majorant. Since $0\le|h(0_{X})|\le A$, $A\ge0$. Put $G(x)=A+A|x|^{2}$; it is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants and Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, hence Borel. For $\rho\in\mathcal{P}_{2}(X)$, additivity of the integral gives $\int_{X}G\,d\rho=A+A\,M_{2}(\rho)<\infty$, since $M_{2}(\rho)<\infty$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space. The function $h$ is continuous, hence Borel, and so is $|h|$ by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space; as $|h|\le G$, monotonicity gives $\int_{X}|h|\,d\rho\le\int_{X}G\,d\rho<\infty$, so $h$ is integrable with respect to $\rho$. This applies to $\rho=\mu$ and to every $\rho=\mu_{j}$.

(ii) Convergence of the majorant. With $\pi_{j}$ as in Step 1, Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §moment gives $|\sqrt{M_{2}(\mu_{j})}-\sqrt{M_{2}(\mu)}|\le\sqrt{I(\pi_{j})}=W_{2}(\mu_{j},\mu)$, so $(\sqrt{M_{2}(\mu_{j})})_{j}$ converges to $\sqrt{M_{2}(\mu)}$ by claim 3 of Order Properties of Limits of Real Sequences; squaring, by claim 2 of Arithmetic of Limits of Real Sequences, $(M_{2}(\mu_{j}))_{j}$ converges to $M_{2}(\mu)$, and by claims 1 and 3 of the same theorem $(\int_{X}G\,d\mu_{j})_{j}$ converges to $\int_{X}G\,d\mu$.

(iii) A lower bound. Let $g:X\to\mathbb{R}$ be continuous, nonnegative and integrable with respect to $\mu$. We show: for every real $\varepsilon>0$ there is $J\in\mathbb{N}$ with $\int_{X}g\,d\mu-\varepsilon<\int_{X}g\,d\mu_{j}$ for every $j\ge J$. For $k\in\mathbb{N}$ let $g_{k}=\min\{g,k\}$; it is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants and claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space, and $0\le g_{k}\le k$, so $g_{k}$ is bounded and Borel, and by Step 1 and the definition of weak convergence $(\int_{X}g_{k}\,d\mu_{j})_{j}$ converges to $\int_{X}g_{k}\,d\mu$. The functions $g_{k}$ are nondecreasing in $k$, and $g_{k}(x)=g(x)$ once $g(x)<k$, which happens for some $k$ by claim 1 of The Archimedean Property of the Real Numbers; so $\sup_{k}g_{k}(x)=g(x)$, and Monotone Convergence Theorem on $(X,\mathcal{B}(X),\mu)$ gives $\int_{X}g\,d\mu=\sup_{k}\int_{X}g_{k}\,d\mu$, a real number. Given $\varepsilon>0$, choose $k$ with $\int_{X}g_{k}\,d\mu>\int_{X}g\,d\mu-\varepsilon/2$ by claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, and then $J$ with $|\int_{X}g_{k}\,d\mu_{j}-\int_{X}g_{k}\,d\mu|<\varepsilon/2$ for $j\ge J$. For such $j$, by monotonicity, $\int_{X}g\,d\mu_{j}\ge\int_{X}g_{k}\,d\mu_{j}>\int_{X}g\,d\mu-\varepsilon$.

(iv) Conclusion. The functions $g_{+}=G-h$ and $g_{-}=G+h$ are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, nonnegative because $|h|\le G$, and integrable with respect to $\mu$ and every $\mu_{j}$, since $0\le g_{\pm}\le2G$. Let $\varepsilon>0$. By (iii) applied to $g_{+}$ and to $g_{-}$, and by (ii) and the definition of the limit, there is $J$ (the largest of the three indices provided) such that for every $j\ge J$

$$\int_{X}g_{+}\,d\mu_{j}>\int_{X}g_{+}\,d\mu-\varepsilon,\qquad\int_{X}g_{-}\,d\mu_{j}>\int_{X}g_{-}\,d\mu-\varepsilon,\qquad\Bigl|\int_{X}G\,d\mu_{j}-\int_{X}G\,d\mu\Bigr|<\varepsilon .$$

For such $j$, by linearity of the integral,

$$\int_{X}h\,d\mu_{j}=\int_{X}G\,d\mu_{j}-\int_{X}g_{+}\,d\mu_{j}<\int_{X}h\,d\mu+2\varepsilon,\qquad\int_{X}h\,d\mu_{j}=\int_{X}g_{-}\,d\mu_{j}-\int_{X}G\,d\mu_{j}>\int_{X}h\,d\mu-2\varepsilon .$$

Thus $|\int_{X}h\,d\mu_{j}-\int_{X}h\,d\mu|<2\varepsilon$ for $j\ge J$; as $\varepsilon>0$ was arbitrary, $(\int_{X}h\,d\mu_{j})_{j}$ converges to $\int_{X}h\,d\mu$.

Step 3 (Claim 3). Assume the hypotheses of claim 3. For $K>0$ write $U_{K}=\{x\in X:K<|x|\}$, the preimage of the open interval $(K,\infty)$ under the Borel map $x\mapsto|x|$, hence a Borel set, and $t_{j}(K)=\int_{U_{K}}|x|^{2}\,\mu_{j}(dx)$.

(a) Truncations. Let $T:X\to X$ satisfy $|Tx-Ty|\le|x-y|$ for all $x,y$ (this holds for $T=\mathrm{id}_{X}$ and, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, for $T=Q_{n}$ with $n\in\mathbb{N}$), and let $R>0$ be real. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §reverse-triangle, $||Tx|-|Ty||\le|Tx-Ty|\le|x-y|$, so $x\mapsto|Tx|$ is Lipschitz, hence continuous; then $u(x)=\min\{|Tx|,R\}$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants and claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space, and $\tau_{T,R}=u\cdot u$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set. Since squaring preserves the order of nonnegative reals, $\tau_{T,R}(x)=\min\{|Tx|^{2},R^{2}\}$, so $0\le\tau_{T,R}\le R^{2}$ and $\tau_{T,R}\le|Tx|^{2}$. Thus $\tau_{T,R}$ is bounded, continuous and Borel, and since $\mu_{j}\Rightarrow\mu$, the sequence $(\int_{X}\tau_{T,R}\,d\mu_{j})_{j}$ converges to $\int_{X}\tau_{T,R}\,d\mu$.

(b) A uniform moment bound. By hypothesis with $\varepsilon=1$ there is $K_{1}>0$ with $t_{j}(K_{1})<1$ for every $j$. Pointwise $|x|^{2}\le K_{1}^{2}+\mathbf{1}_{U_{K_{1}}}(x)|x|^{2}$, so by additivity and monotonicity $M_{2}(\mu_{j})\le K_{1}^{2}+t_{j}(K_{1})\le K_{1}^{2}+1$ for every $j$.

(c) $\mu\in\mathcal{P}_{2}(X)$. For $r\in\mathbb{N}$ put $\tau_{r}=\tau_{\mathrm{id}_{X},r}$, so $\tau_{r}(x)=\min\{|x|^{2},r^{2}\}$. By monotonicity and (b), $\int_{X}\tau_{r}\,d\mu_{j}\le M_{2}(\mu_{j})\le K_{1}^{2}+1$ for every $j$, so by (a) and claim 1 of Order Properties of Limits of Real Sequences (compared with the constant sequence $K_{1}^{2}+1$), $\int_{X}\tau_{r}\,d\mu\le K_{1}^{2}+1$. The functions $\tau_{r}$ are nondecreasing in $r$, and $\tau_{r}(x)=|x|^{2}$ once $|x|<r$, which happens for some $r$ by claim 1 of The Archimedean Property of the Real Numbers; so $\sup_{r}\tau_{r}(x)=|x|^{2}$, and Monotone Convergence Theorem on $(X,\mathcal{B}(X),\mu)$ gives $M_{2}(\mu)=\sup_{r}\int_{X}\tau_{r}\,d\mu\le K_{1}^{2}+1<\infty$. As $\mu\in\mathcal{P}(X)$, $\mu\in\mathcal{P}_{2}(X)$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space.

(d) A uniform tail estimate. Let $n\in\mathbb{N}$ and $K>0$. For every $x\in X$, $|Q_{n}x|^{2}\le\tau_{Q_{n},K}(x)+\mathbf{1}_{U_{K}}(x)|x|^{2}$: if $|x|\le K$ then $|Q_{n}x|\le|x|\le K$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so $|Q_{n}x|^{2}=\tau_{Q_{n},K}(x)$; if $K<|x|$ then $|Q_{n}x|^{2}\le|x|^{2}=\mathbf{1}_{U_{K}}(x)|x|^{2}$. By additivity and monotonicity, $\int_{X}|Q_{n}x|^{2}\,\mu_{j}(dx)\le\int_{X}\tau_{Q_{n},K}\,d\mu_{j}+t_{j}(K)$ for every $j$.

(e) The finite-dimensional term. Fix $n\in\mathbb{N}$. By The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections §projected, $(p_{n})_{\#}\mu_{j}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ for every $j$, and $(p_{n})_{\#}\mu\in\mathcal{P}(\mathbb{R}^{n})$, the Borel $\sigma$-algebra of $(\mathbb{R}^{n},d_{E})$ being $\mathcal{B}(\mathbb{R}^{n})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. The map $p_{n}$ is Lipschitz from $(X,d)$ to $(\mathbb{R}^{n},d_{E})$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence continuous, so Weak Convergence of Finite Borel Measures is Preserved by Continuous Maps between Metric Spaces §weak, applied with $(Y,d_{Y})=(X,d)$, $(Z,d_{Z})=(\mathbb{R}^{n},d_{E})$ and $T=p_{n}$, gives $(p_{n})_{\#}\mu_{j}\Rightarrow(p_{n})_{\#}\mu$ for the Euclidean distance. Let $\varepsilon'>0$ and let $K>0$ be given by the hypothesis of claim 3 for $\varepsilon'$. The function $y\mapsto\mathbf{1}_{\{K<\lVert y\rVert\}}(y)\lVert y\rVert^{2}$ on $\mathbb{R}^{n}$ is Borel, by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and since $\{y:K<\lVert y\rVert\}$ is the preimage of $(K,\infty)$ under a Borel map; and for $x\in X$, $\lVert p_{n}(x)\rVert=|P_{n}x|\le|x|$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so $\mathbf{1}_{\{K<\lVert p_{n}(x)\rVert\}}\lVert p_{n}(x)\rVert^{2}\le\mathbf{1}_{U_{K}}(x)|x|^{2}$. By change of variables (claim 2 of Image Measures, Measures with Densities, and Change of Variables) and monotonicity,

$$\int_{\{y:\,K<\lVert y\rVert\}}\lVert y\rVert^{2}\,(p_{n})_{\#}\mu_{j}(dy)=\int_{X}\mathbf{1}_{\{K<\lVert p_{n}(x)\rVert\}}\lVert p_{n}(x)\rVert^{2}\,\mu_{j}(dx)\le t_{j}(K)<\varepsilon'$$

for every $j$. Hence Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §convergence, applied with $m=n$ to the sequence $((p_{n})_{\#}\mu_{j})_{j}$ and the measure $(p_{n})_{\#}\mu$, shows that $(W_{2}^{(n)}((p_{n})_{\#}\mu_{j},(p_{n})_{\#}\mu))_{j}$ converges to $0$, where $W_{2}^{(n)}$ is the quadratic Wasserstein distance on $\mathcal{P}_{2}(\mathbb{R}^{n})$, which is the distance $W_{2}$ of that lemma by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions.

(f) Conclusion. Let $\varepsilon>0$ and put $\eta=\varepsilon/4$. The choices are made in the following order. First, by hypothesis (with $\eta^{2}$ in place of $\varepsilon$), choose $K>0$ with $t_{j}(K)<\eta^{2}$ for every $j$. Second, since $\mu\in\mathcal{P}_{2}(X)$ by (c), The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections §tail gives $n\in\mathbb{N}$ with $\int_{X}|Q_{n}x|^{2}\,\mu(dx)<\eta^{2}$; then $\int_{X}\tau_{Q_{n},K}\,d\mu\le\int_{X}|Q_{n}x|^{2}\,\mu(dx)<\eta^{2}$ by monotonicity. Third, by (a) choose $J_{1}$ with $\int_{X}\tau_{Q_{n},K}\,d\mu_{j}<\int_{X}\tau_{Q_{n},K}\,d\mu+\eta^{2}<2\eta^{2}$ for $j\ge J_{1}$. Fourth, by (e) choose $J_{2}$ with $W_{2}^{(n)}((p_{n})_{\#}\mu_{j},(p_{n})_{\#}\mu)<\eta$ for $j\ge J_{2}$. Let $j\ge\max\{J_{1},J_{2}\}$. By (d), $\int_{X}|Q_{n}x|^{2}\,\mu_{j}(dx)<2\eta^{2}+\eta^{2}=3\eta^{2}<(2\eta)^{2}$. By The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections §projected, the measures $\mu_{j},\mu,(P_{n})_{\#}\mu_{j},(P_{n})_{\#}\mu$ lie in $\mathcal{P}_{2}(X)$; by The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections §tail, $W_{2}((P_{n})_{\#}\mu_{j},\mu_{j})<2\eta$ and $W_{2}((P_{n})_{\#}\mu,\mu)<\eta$; and by The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections §isometry, $W_{2}((P_{n})_{\#}\mu_{j},(P_{n})_{\#}\mu)=W_{2}^{(n)}((p_{n})_{\#}\mu_{j},(p_{n})_{\#}\mu)<\eta$. Applying The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §triangle twice and The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §symmetry,

$$W_{2}(\mu_{j},\mu)\le W_{2}(\mu_{j},(P_{n})_{\#}\mu_{j})+W_{2}((P_{n})_{\#}\mu_{j},(P_{n})_{\#}\mu)+W_{2}((P_{n})_{\#}\mu,\mu)<2\eta+\eta+\eta=\varepsilon .$$

Since $W_{2}(\mu_{j},\mu)\ge0$, this shows that $(W_{2}(\mu_{j},\mu))_{j}$ converges to $0$, and together with (c) proves claim 3. The argument used only that $(\mu_{j})_{j}$ is a sequence in $\mathcal{P}_{2}(X)$ converging weakly to $\mu\in\mathcal{P}(X)$ and satisfying the uniform tail condition, so it applies to every such sequence.

Step 4 (Claim 4). Each $\mu_{j}$ is a Borel measure on $(X,d)$ with $\mu_{j}(X)=1$, and the sequence is tight, so Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence gives a strictly increasing sequence $(j_{i})_{i\in\mathbb{N}}$ in $\mathbb{N}$ and a Borel measure $\mu$ on $(X,d)$ with $\mu(X)=1$, that is, $\mu\in\mathcal{P}(X)$, such that $\mu_{j_{i}}\Rightarrow\mu$. The sequence $(\mu_{j_{i}})_{i}$ lies in $\mathcal{P}_{2}(X)$, and for every $\varepsilon>0$ the $K$ given by the hypothesis satisfies $\int_{U_{K}}|x|^{2}\,\mu_{j_{i}}(dx)<\varepsilon$ for every $i$, since the bound holds for every index. By Step 3 applied to the sequence $(\mu_{j_{i}})_{i}$ and the measure $\mu$, $\mu\in\mathcal{P}_{2}(X)$ and $(W_{2}(\mu_{j_{i}},\mu))_{i}$ converges to $0$. $\blacksquare$