Proof of Completeness of the Quadratic Wasserstein Space and of the Noise Wasserstein Space over a Hilbert Space

Each result cited is universally quantified over the data in its own statement.

Throughout, the notation is that of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, whose clause Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background puts the notation of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation in force; couplings are written $\gamma_{k}$, the letters $\pi_{1},\pi_{2}$ being the coordinate maps of $X\times X$. Square roots are the nonnegative ones of Existence and Uniqueness of the Nonnegative Square Root; the monotonicity of $s\mapsto\sqrt{s}$ on $[0,\infty)$ follows from claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and the identity $\sqrt{s^{2}}=s$ for $s\ge0$ from the uniqueness in that theorem, both results being in force by The Real Numbers: Standing Notation and Background §background. A strictly increasing sequence in $\mathbb{N}$ composed with a strictly increasing sequence in $\mathbb{N}$ is strictly increasing by claim 2 of A Subsequence of a Subsequence is a Subsequence.

Preliminaries. We use three elementary facts.

Claim (subsequences). Let $Y$ be $X$ or $X\times X$, let $\kappa_{n},\kappa\in\mathcal{P}(Y)$ for $n\in\mathbb{N}$ with $\kappa_{n}\Rightarrow\kappa$, and let $(n_{k})_{k\in\mathbb{N}}$ be strictly increasing in $\mathbb{N}$. Then $\kappa_{n_{k}}\Rightarrow\kappa$. Indeed, fix a bounded continuous $f:Y\to\mathbb{R}$. By Weak Convergence of Finite Borel Measures on a Metric Space the real sequence $(\int_{Y}f\,d\kappa_{n})_{n}$ converges to $\int_{Y}f\,d\kappa$ in the sense of Limit of a Sequence of Real Numbers, which is convergence in the real line of The Absolute Value Metric on the Real Line in the sense of Convergent Sequence in a Metric Space, both definitions requiring $|\int_{Y}f\,d\kappa_{n}-\int_{Y}f\,d\kappa|<\varepsilon$ for all large $n$. By A Subsequence of a Convergent Sequence Has the Same Limit the subsequence $(\int_{Y}f\,d\kappa_{n_{k}})_{k}$ converges to the same limit; as $f$ was arbitrary, $\kappa_{n_{k}}\Rightarrow\kappa$. Likewise, a constant sequence with all terms $\kappa\in\mathcal{P}(Y)$ converges weakly to $\kappa$, its integrals forming a constant sequence, which converges by Constant Sequences and Index-Shifted Sequences of Real Numbers §constant.

Claim (tight subfamilies). A subset of a tight set of Borel measures on $(Y,d)$ is tight, the compact set supplied for the larger set by Tight Family of Borel Measures on a Metric Space §tight serving for the subset; so a sequence whose terms lie in a tight set is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence.

Claim (Prokhorov). $(X,d)$ and $(X\times X,d)$ are metric spaces by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs, the members of $\mathcal{P}(X)$ and $\mathcal{P}(X\times X)$ are Borel measures of total mass $1$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, and $\Rightarrow$ is the weak convergence of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §weak. Hence, by Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence, every tight sequence in $\mathcal{P}(X)$ or in $\mathcal{P}(X\times X)$ has a subsequence converging weakly to a member of $\mathcal{P}(X)$, respectively $\mathcal{P}(X\times X)$.

Step 1 (claim 1). Let $(\mu_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in $(\mathcal{P}_{2}(X),W_{2})$, in the sense of Cauchy Sequence in a Metric Space. By Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight §tight it is tight in $(X,d)$, so by Claim (Prokhorov) there are a strictly increasing $(n_{k})_{k\in\mathbb{N}}$ and $\mu\in\mathcal{P}(X)$ with $\mu_{n_{k}}\Rightarrow\mu$.

Claim (tail estimate). For every $\varepsilon>0$ there is $N\in\mathbb{N}$ such that, for every $m\ge N$, $\mu\in\mathcal{P}_{2}(X)$ and $W_{2}(\mu_{m},\mu)\le\varepsilon/2$.

Proof of the claim. Fix $\varepsilon>0$, and by Cauchy Sequence in a Metric Space choose $N$ with $W_{2}(\mu_{m},\mu_{l})<\varepsilon/2$ for all $m,l\ge N$. Fix $m\ge N$. For $k\in\mathbb{N}$ put $r_{k}=n_{N+k}$; the sequence $(r_{k})_{k}$ is strictly increasing, $r_{k}\ge N+k\ge N$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, and $\mu_{r_{k}}\Rightarrow\mu$ by Claim (subsequences), being a subsequence of $(\mu_{n_{k}})_{k}$. For each $k$ choose, by Existence of an Optimal Coupling of Two Borel Probability Measures with Finite Second Moment on a Hilbert Space §existence, a coupling $\gamma_{k}\in\Pi(\mu_{m},\mu_{r_{k}})$ with $I(\gamma_{k})=W_{2}(\mu_{m},\mu_{r_{k}})^{2}$ (a countable sequence of choices, in the order of $k$); then $0\le I(\gamma_{k})<\varepsilon^{2}/4$ for every $k$, so $(I(\gamma_{k}))_{k}$ is a bounded sequence of real numbers.

The set $\{\mu_{m}\}$ is tight by Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight §tight, $(X,d)$ being complete and separable by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space; the set $\{\mu_{r_{k}}:k\in\mathbb{N}\}$ is tight by Claim (tight subfamilies), being contained in the tight set $\{\mu_{n}:n\in\mathbb{N}\}$. By Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight the set of all couplings of $\mu_{m}$ with some $\mu_{r_{k}}$ is tight in $X\times X$; it contains every $\gamma_{k}$, so $(\gamma_{k})_{k}$ is tight by Claim (tight subfamilies). By Claim (Prokhorov) there are a strictly increasing $(k_{i})_{i\in\mathbb{N}}$ and $\gamma\in\mathcal{P}(X\times X)$ with $\gamma_{k_{i}}\Rightarrow\gamma$. By Claim (subsequences), the constant sequence with terms $\mu_{m}$ converges weakly to $\mu_{m}$ and $\mu_{r_{k_{i}}}\Rightarrow\mu$. Now Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §closed, applied with the marginals $\mu_{m}$ and $\mu_{r_{k_{i}}}$ and the couplings $\gamma_{k_{i}}\in\Pi(\mu_{m},\mu_{r_{k_{i}}})$, gives $\gamma\in\Pi(\mu_{m},\mu)$. Every $I(\gamma_{k_{i}})$ is a real number and $(I(\gamma_{k_{i}}))_{i}$ is bounded, so Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §lsc gives $I(\gamma)<\infty$ and $I(\gamma)\le\liminf_{i}I(\gamma_{k_{i}})$; and $\liminf_{i}I(\gamma_{k_{i}})\le\limsup_{i}I(\gamma_{k_{i}})\le\varepsilon^{2}/4$ by claims 1 and 2 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence.

Since $\mu_{m}\in\mathcal{P}_{2}(X)$, $\gamma\in\Pi(\mu_{m},\mu)$ and $I(\gamma)<\infty$, the converse part 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 §cost-finite gives $\mu\in\mathcal{P}_{2}(X)$. Then $W_{2}(\mu_{m},\mu)$ is defined and $W_{2}(\mu_{m},\mu)^{2}\le I(\gamma)\le\varepsilon^{2}/4$ by The Quadratic Wasserstein Distance on a Hilbert Space §distance; taking square roots, $W_{2}(\mu_{m},\mu)\le\varepsilon/2$. This proves the claim.

By the claim (with $\varepsilon=1$, say) $\mu\in\mathcal{P}_{2}(X)$, and for every $\varepsilon>0$ there is $N$ with $W_{2}(\mu_{m},\mu)\le\varepsilon/2<\varepsilon$ for all $m\ge N$. By Convergent Sequence in a Metric Space, $(\mu_{n})_{n}$ converges to $\mu\in\mathcal{P}_{2}(X)$ in the metric space $(\mathcal{P}_{2}(X),W_{2})$ of The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §metric. As the Cauchy sequence was arbitrary, $(\mathcal{P}_{2}(X),W_{2})$ is complete by Complete Metric Space.

Step 2 (claim 2). Let $(\mu_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric. Every $\mu_{n}$ belongs to $\mathcal{P}_{2}(X)$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments.

Claim (W_2-Cauchy). $(\mu_{n})_{n}$ is a Cauchy sequence in $(\mathcal{P}_{2}(X),W_{2})$. Indeed, $\bar{a}>0$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, so $\sqrt{\bar{a}}>0$. Given $\varepsilon>0$, choose by Cauchy Sequence in a Metric Space an $N$ with $W_{a}(\mu_{m},\mu_{l})<\varepsilon/\sqrt{\bar{a}}$ for all $m,l\ge N$; then The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §comparison gives $W_{2}(\mu_{m},\mu_{l})\le\sqrt{\bar{a}}\,W_{a}(\mu_{m},\mu_{l})<\varepsilon$ for all $m,l\ge N$.

By the claim and Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight §tight, $(\mu_{n})_{n}$ is tight in $(X,d)$, and by Claim (Prokhorov) there are a strictly increasing $(n_{k})_{k\in\mathbb{N}}$ and $\mu\in\mathcal{P}(X)$ with $\mu_{n_{k}}\Rightarrow\mu$.

Fix $\varepsilon>0$, choose by Cauchy Sequence in a Metric Space an $N$ with $W_{a}(\mu_{m},\mu_{l})<\varepsilon/2$ for all $m,l\ge N$, and fix $m\ge N$. For $k\in\mathbb{N}$ put $r_{k}=n_{N+k}$, so that $(r_{k})_{k}$ is strictly increasing, $r_{k}\ge N$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, and $\mu_{r_{k}}\Rightarrow\mu$ by Claim (subsequences); the constant sequence with terms $\mu_{m}$ converges weakly to $\mu_{m}$ by the same claim. All $\mu_{m}$ and $\mu_{r_{k}}$ lie in $\mathcal{P}^{a}_{\rho}$, and $0\le W_{a}(\mu_{m},\mu_{r_{k}})<\varepsilon/2$ for every $k$, so the sequence $(W_{a}(\mu_{m},\mu_{r_{k}}))_{k}$ is bounded. By The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §lsc, applied with the constant sequence $\mu_{m}$ in the role of $(\mu_{j})$, the sequence $(\mu_{r_{k}})_{k}$ in the role of $(\nu_{j})$, and the limits $\mu_{m}$ and $\mu$, the pair $(\mu_{m},\mu)$ is noise-connected and $W_{a}(\mu_{m},\mu)\le\liminf_{k}W_{a}(\mu_{m},\mu_{r_{k}})$, which is at most $\limsup_{k}W_{a}(\mu_{m},\mu_{r_{k}})\le\varepsilon/2$ by claims 1 and 2 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence. Since $\mu_{m}\in\mathcal{P}^{a}_{\rho}$, The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §closed-under-limit gives $\mu\in\mathcal{P}^{a}_{\rho}$.

Thus $\mu\in\mathcal{P}^{a}_{\rho}$, and for every $\varepsilon>0$ there is $N$ with $W_{a}(\mu_{m},\mu)\le\varepsilon/2<\varepsilon$ for every $m\ge N$, where $W_{a}(\mu_{m},\mu)$ is the value of the metric of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric at $(\mu_{m},\mu)$. By Convergent Sequence in a Metric Space, $(\mu_{n})_{n}$ converges to $\mu\in\mathcal{P}^{a}_{\rho}$ in $(\mathcal{P}^{a}_{\rho},W_{a})$, and as the Cauchy sequence was arbitrary, $(\mathcal{P}^{a}_{\rho},W_{a})$ is complete by Complete Metric Space.