Proof of A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function

Each result cited is universally quantified over the data in its own statement. Elementary arithmetic and order of real numbers, finite sums of real numbers, square roots of nonnegative reals and the absolute value are used freely; they are carried by The Real Numbers: Standing Notation and Background §background. For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, as in The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings, and $m(\rho)\in\mathbb{R}^{d}$ is the mean of $\rho$, that of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean. For $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the second moment $M_{2}(\rho)$ is a nonnegative real number, being the integral of a nonnegative function (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment) and finite. Since $0\le M_{2}(\mu_{1})\le B$, the number $B$ is nonnegative, and for $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ we put

$$R_{\rho}=\sqrt{M_{2}(\rho)}+\sqrt{B}\ \ge0 .$$

For $K\in\mathbb{N}$ let $\sigma_{K}=\sum_{k=1}^{K}c_{k}$ be the $K$-th partial sum of $(c_{k})$ (Series of Real Numbers §partial-sums) and $r_{K}=c-\sigma_{K}$. Since $0<c_{k}$, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates gives $0\le\sigma_{K}\le c$, so $0\le r_{K}$; also $0<c_{1}=\sigma_{1}\le c$, so $c$ is positive. By Series of Real Numbers §convergent the sequence $(\sigma_{K})_{K\in\mathbb{N}}$ converges to $c$, so by Arithmetic of Limits of Real Sequences §scalar (its statement on differences, applied to the constant sequence $c$ minus $(\sigma_{K})$) the sequence $(r_{K})_{K\in\mathbb{N}}$ converges to $0$; by Arithmetic of Limits of Real Sequences §products and Arithmetic of Limits of Real Sequences §scalar so does $(\lambda\,r_{K}^{2})_{K\in\mathbb{N}}$ and $(\lambda\,r_{K})_{K\in\mathbb{N}}$ for every real $\lambda$. Hence for every positive real $\delta$ and every real $\lambda$ there is $K\in\mathbb{N}$ with $\lambda r_{K}<\delta$, and likewise with $\lambda r_{K}^{2}<\delta$.

Step 1 (Elementary bounds). (i) For $\rho,\rho'\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and every $\pi\in\Pi(\rho,\rho')$, $W_{2}(\rho,\rho')\le\sqrt{I(\pi)}$, by The Quadratic Wasserstein Distance on Euclidean Space §distance and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

(ii) For $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $k\in\mathbb{N}$, $W_{2}(\rho,\mu_{k})\le R_{\rho}$. Indeed, by Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment there is $\pi\in\Pi(\rho,\mu_{k})$ with $I(\pi)=W_{2}(\rho,\mu_{k})^{2}$. By The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance §fields, read with $m=d$, the coordinate projections define elements $\mathrm{pr}_{1},\mathrm{pr}_{2}$ of the real inner product space $L^{2}(\pi;\mathbb{R}^{d})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, read with $q=d+d$ and $r=d$) with $\lVert\mathrm{pr}_{1}\rVert_{\pi}^{2}=M_{2}(\rho)$, $\lVert\mathrm{pr}_{2}\rVert_{\pi}^{2}=M_{2}(\mu_{k})$ and $\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert_{\pi}^{2}=I(\pi)$. All of these norms and $W_{2}(\rho,\mu_{k})$ are nonnegative, so by uniqueness of nonnegative square roots $\lVert\mathrm{pr}_{1}\rVert_{\pi}=\sqrt{M_{2}(\rho)}$, $\lVert\mathrm{pr}_{2}\rVert_{\pi}=\sqrt{M_{2}(\mu_{k})}$ and $\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert_{\pi}=W_{2}(\rho,\mu_{k})$. The triangle inequality of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle in $L^{2}(\pi;\mathbb{R}^{d})$ gives $W_{2}(\rho,\mu_{k})\le\sqrt{M_{2}(\rho)}+\sqrt{M_{2}(\mu_{k})}$, and $\sqrt{M_{2}(\mu_{k})}\le\sqrt{B}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, since $M_{2}(\mu_{k})\le B$.

(iii) Let $\rho,\rho'\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfy $W_{2}(\rho,\rho')\le1$, and let $k\in\mathbb{N}$. Then

$$W_{2}(\rho',\mu_{k})\le R_{\rho}+1\qquad\text{and}\qquad\bigl|W_{2}(\rho',\mu_{k})^{2}-W_{2}(\rho,\mu_{k})^{2}\bigr|\le(2R_{\rho}+1)\,W_{2}(\rho,\rho').$$

Indeed, by the triangle inequality and symmetry of $W_{2}$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry), $W_{2}(\rho',\mu_{k})\le W_{2}(\rho',\rho)+W_{2}(\rho,\mu_{k})$ and $W_{2}(\rho,\mu_{k})\le W_{2}(\rho,\rho')+W_{2}(\rho',\mu_{k})$. The first of these and (ii) give $W_{2}(\rho',\mu_{k})\le1+R_{\rho}$, and together they give $|W_{2}(\rho',\mu_{k})-W_{2}(\rho,\mu_{k})|\le W_{2}(\rho,\rho')$. Writing $s=W_{2}(\rho',\mu_{k})$ and $t=W_{2}(\rho,\mu_{k})$, both nonnegative, $|s^{2}-t^{2}|=|s-t|\,(s+t)\le W_{2}(\rho,\rho')\,(R_{\rho}+1+R_{\rho})$.

Step 2 (Two facts on series). (T) Tail estimate. Let $E$ be a real inner product space with norm $|\cdot|$, let $(x_{k})_{k\in\mathbb{N}}$ be a sequence in $E$ whose series converges (Series in a Real Inner Product Space §convergent), with partial sums $s_{n}$ and sum $x$, and let $(b_{k})_{k\in\mathbb{N}}$ be real numbers with $|x_{k}|\le b_{k}$ for every $k$ whose series converges, with partial sums $\beta_{n}$ and sum $b$. Then $|x-s_{K}|\le b-\beta_{K}$ for every $K\in\mathbb{N}$. Proof: fix $K$. We show $|s_{K+n}-s_{K}|\le\beta_{K+n}-\beta_{K}$ for every $n\in\mathbb{N}$ by induction on $n$. By the recursive definition of finite sums (Finite Sum Notation in a Vector Space, and The Real Numbers: Standing Notation and Background §naturals for real numbers), $s_{K+1}-s_{K}=x_{K+1}$ and $s_{K+n+1}-s_{K}=(s_{K+n}-s_{K})+x_{K+n+1}$, and similarly for $\beta$; so the case $n=1$ is the hypothesis $|x_{K+1}|\le b_{K+1}$, and the inductive step follows from the triangle inequality of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and the hypothesis $|x_{K+n+1}|\le b_{K+n+1}$. Since $0\le|x_{k}|\le b_{k}$, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates gives $\beta_{K+n}\le b$, whence $|s_{K+n}-s_{K}|\le b-\beta_{K}$ for every $n$. The sequence $(s_{K+n})_{n\in\mathbb{N}}$ converges to $x$, because $K+n\ge n$ and $(s_{n})$ converges to $x$; so $(s_{K+n}-s_{K})_{n}$ converges to $x-s_{K}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits, and $(|s_{K+n}-s_{K}|)_{n}$ converges to $|x-s_{K}|$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity. Claim 1 of Order Properties of Limits of Real Sequences, applied with the constant sequence $b-\beta_{K}$, gives $|x-s_{K}|\le b-\beta_{K}$. We use (T) for $E=L^{2}(\rho;\mathbb{R}^{d})$ and for $E=\mathbb{R}$, the latter being the real inner product space of Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces §real-line with norm the absolute value, in which convergence of a series and its partial sums and sum agree with those of Series of Real Numbers by Elementary Properties of Series in a Real Inner Product Space §real-line.

(S) Signed comparison. Let $(a_{k})$ and $(b_{k})$ be real sequences with $|a_{k}|\le b_{k}$ for every $k$ such that $\sum_{k=1}^{\infty}b_{k}$ converges with sum $b$. Then $\sum_{k=1}^{\infty}a_{k}$ converges and $|\sum_{k=1}^{\infty}a_{k}|\le b$. Proof: $0\le a_{k}+b_{k}\le2b_{k}$, and $\sum_{k}2b_{k}$ converges by Elementary Properties of Series of Real Numbers §linearity, so $\sum_{k}(a_{k}+b_{k})$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison; then $\sum_{k}a_{k}=\sum_{k}\bigl((a_{k}+b_{k})+(-1)b_{k}\bigr)$ converges by Elementary Properties of Series of Real Numbers §linearity. As $a_{k}\le b_{k}$ and $-a_{k}\le b_{k}$, and $\sum_{k}(-a_{k})=-\sum_{k}a_{k}$ by the same clause, Elementary Properties of Series of Real Numbers §order gives $\sum_{k}a_{k}\le b$ and $-\sum_{k}a_{k}\le b$.

Step 3 (Clause 1). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For every $k$, $0\le W_{2}(\mu,\mu_{k})\le R_{\mu}$ by Step 1(ii), hence $0\le W_{2}(\mu,\mu_{k})^{2}\le R_{\mu}^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound, read with $(c_{k})$ in place of its $(\mu_{k})$, with $M=R_{\mu}$ and $w_{k}=W_{2}(\mu,\mu_{k})$, and again with $M=R_{\mu}^{2}$ and $w_{k}=W_{2}(\mu,\mu_{k})^{2}$, both series of clause 1 converge. So $\psi$ is well defined. For $K\in\mathbb{N}$ let $\psi_{K}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $\psi_{K}(\rho)=\sum_{k=1}^{K}c_{k}W_{2}(\rho,\mu_{k})^{2}$, be the $K$-th partial sum of the series defining $\psi(\rho)$.

Step 4 (Property (a): continuity). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\varepsilon\in\mathbb{R}$ be positive. Put $\delta=\min\{1,\varepsilon\,(c\,(2R_{\mu}+1))^{-1}\}$, which is positive. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfy $W_{2}(\mu,\nu)<\delta$, and put $d_{k}=W_{2}(\nu,\mu_{k})^{2}-W_{2}(\mu,\mu_{k})^{2}$. By Elementary Properties of Series of Real Numbers §linearity the series $\sum_{k}c_{k}d_{k}$ converges with sum $\psi(\nu)-\psi(\mu)$. Since $W_{2}(\mu,\nu)\le1$, Step 1(iii) gives $|c_{k}d_{k}|\le c_{k}(2R_{\mu}+1)W_{2}(\mu,\nu)$, and the series of the right-hand sides converges with sum $c\,(2R_{\mu}+1)W_{2}(\mu,\nu)$ by Elementary Properties of Series of Real Numbers §linearity. By Step 2(S),

$$|\psi(\nu)-\psi(\mu)|\le c\,(2R_{\mu}+1)\,W_{2}(\mu,\nu)<c\,(2R_{\mu}+1)\,\delta\le\varepsilon .$$

So $\psi$ is continuous at $\mu$, for the metric $W_{2}$ and the absolute-value metric; as $\mu$ was arbitrary, property (a) holds.

Step 5 (The candidate gradients). Let $\rho\in Q$. By The Map Property of a Set of Probability Measures §map-property, for every $k$ the pair $(\rho,\mu_{k})$ is uniquely mapped; let $S^{\rho}_{k}$ be any optimal map from $\rho$ to $\mu_{k}$; then $\mathrm{id}-S^{\rho}_{k}\in T_{\rho}$. Put $x^{\rho}_{k}=2c_{k}(\mathrm{id}-S^{\rho}_{k})\in L^{2}(\rho;\mathbb{R}^{d})$ and let $s^{\rho}_{K}=\sum_{k=1}^{K}x^{\rho}_{k}$ be its partial sums (Series in a Real Inner Product Space §partial-sums). Recall that $L^{2}(\rho;\mathbb{R}^{d})$ is a real Hilbert space (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields).

(i) $\lVert x^{\rho}_{k}\rVert_{\rho}=2c_{k}W_{2}(\rho,\mu_{k})$. Indeed $\lVert\mathrm{id}-S^{\rho}_{k}\rVert_{\rho}^{2}=W_{2}(\rho,\mu_{k})^{2}$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §cost, so $\lVert\mathrm{id}-S^{\rho}_{k}\rVert_{\rho}=W_{2}(\rho,\mu_{k})$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both being nonnegative; and $\lVert\lambda\xi\rVert_{\rho}=|\lambda|\,\lVert\xi\rVert_{\rho}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric (with the zero vector as second point), with $|2c_{k}|=2c_{k}$.

(ii) The series $\sum_{k}x^{\rho}_{k}$ converges in $L^{2}(\rho;\mathbb{R}^{d})$; we write $\eta_{\rho}$ for its sum. Moreover $\lVert\eta_{\rho}\rVert_{\rho}\le2\sum_{k=1}^{\infty}c_{k}W_{2}(\rho,\mu_{k})$. Indeed, by (i), Step 3 and Elementary Properties of Series of Real Numbers §linearity the series $\sum_{k}\lVert x^{\rho}_{k}\rVert_{\rho}$ converges with sum $2\sum_{k}c_{k}W_{2}(\rho,\mu_{k})$, so $\sum_{k}x^{\rho}_{k}$ converges absolutely, and Elementary Properties of Series in a Real Inner Product Space §absolute gives both assertions.

(iii) If $M\in\mathbb{R}$ is nonnegative and $W_{2}(\rho,\mu_{k})\le M$ for every $k$, then $\lVert\eta_{\rho}-s^{\rho}_{K}\rVert_{\rho}\le2M\,r_{K}$ for every $K\in\mathbb{N}$. Indeed $\lVert x^{\rho}_{k}\rVert_{\rho}\le2Mc_{k}$ by (i), the series $\sum_{k}2Mc_{k}$ converges with sum $2Mc$ by Elementary Properties of Series of Real Numbers §linearity, its $K$-th partial sum is $2M\sigma_{K}$, and Step 2(T) in $E=L^{2}(\rho;\mathbb{R}^{d})$ gives $\lVert\eta_{\rho}-s^{\rho}_{K}\rVert_{\rho}\le2Mc-2M\sigma_{K}=2Mr_{K}$.

(iv) $\eta_{\rho}\in T_{\rho}$. By Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed, $T_{\rho}$ is a closed linear subspace of $L^{2}(\rho;\mathbb{R}^{d})$. Each $x^{\rho}_{k}$ is a real multiple of $\mathrm{id}-S^{\rho}_{k}\in T_{\rho}$, hence lies in $T_{\rho}$ (Linear Subspace); since $s^{\rho}_{1}=x^{\rho}_{1}$ and $s^{\rho}_{K+1}=s^{\rho}_{K}+x^{\rho}_{K+1}$ (Finite Sum Notation in a Vector Space), induction on $K$ gives $s^{\rho}_{K}\in T_{\rho}$ for every $K$. The sequence $(s^{\rho}_{K})$ converges to $\eta_{\rho}$ (Series in a Real Inner Product Space §convergent) and $T_{\rho}$ is closed in the sense of Real Hilbert Space §topology, so $\eta_{\rho}\in T_{\rho}$ by Sequential Characterization of Closed Subsets of a Metric Space.

(v) For every $K\in\mathbb{N}$, $\psi_{K}$ is an intrinsic test function on $Q$ and $\nabla\psi_{K}(\rho)=s^{\rho}_{K}$ for every $\rho\in Q$. Indeed, for each $k$ the function $\varphi_{k}(\rho)=W_{2}(\rho,\mu_{k})^{2}$ is an intrinsic test function on $Q$ with $\nabla\varphi_{k}(\rho)=2(\mathrm{id}-S^{\rho}_{k})$ for $\rho\in Q$, by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance read with $\nu_{0}=\mu_{k}$. Now $\psi_{1}=c_{1}\varphi_{1}+0\,\varphi_{1}$ and $\psi_{K+1}=1\,\psi_{K}+c_{K+1}\varphi_{K+1}$ by the recursive definition of finite sums, so induction on $K$ with Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear shows that $\psi_{K}$ is an intrinsic test function on $Q$ with $\nabla\psi_{K}(\rho)=\sum_{k=1}^{K}c_{k}\cdot2(\mathrm{id}-S^{\rho}_{k})=s^{\rho}_{K}$, the vector-space axioms of $L^{2}(\rho;\mathbb{R}^{d})$ giving $c_{k}\cdot2(\mathrm{id}-S^{\rho}_{k})=x^{\rho}_{k}$.

Step 6 (Property (b), and clause 3). Let $\mu\in Q$, let $S_{k}=S^{\mu}_{k}$ be the optimal maps chosen in Step 5 (an arbitrary choice), and write $\eta=\eta_{\mu}$ and $s_{K}=s^{\mu}_{K}$. We show that $\psi$ is differentiable along couplings at $\mu$ with gradient $\eta$. Let $\varepsilon\in\mathbb{R}$ be positive. The order of choice is: first $K$, then $\theta_{1}$, then $\theta$. Choose $K\in\mathbb{N}$ with $(4R_{\mu}+1)\,r_{K}<\varepsilon\cdot2^{-1}$, which is possible by the opening paragraph. By Step 5(v) and property (b) for $\psi_{K}$, the function $\psi_{K}$ is differentiable along couplings at $\mu$ with gradient $s_{K}$ (Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §gradient); choose a positive $\theta_{1}$ as in Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable for $\psi_{K}$, $s_{K}$ and $\varepsilon\cdot2^{-1}$, and put $\theta=\min\{\theta_{1},1\}$, which is positive.

Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ satisfy $I(\pi)<\theta^{2}$. As $\theta\le\theta_{1}$ and $\theta\le1$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $I(\pi)<\theta_{1}^{2}$ and $I(\pi)<1$, so $\sqrt{I(\pi)}<1$ by claim 1 of that lemma, and $W_{2}(\mu,\nu)\le\sqrt{I(\pi)}<1$ by Step 1(i). Put $d_{k}=W_{2}(\nu,\mu_{k})^{2}-W_{2}(\mu,\mu_{k})^{2}$.

First, the real tail. By Elementary Properties of Series of Real Numbers §linearity the series $\sum_{k}c_{k}d_{k}$ converges with sum $\psi(\nu)-\psi(\mu)$, and its $K$-th partial sum is $\psi_{K}(\nu)-\psi_{K}(\mu)$. By Step 1(iii), applied with $\rho=\mu$ and $\rho'=\nu$ (admissible as $W_{2}(\mu,\nu)\le1$), and Step 1(i), $|d_{k}|\le(2R_{\mu}+1)W_{2}(\mu,\nu)\le(2R_{\mu}+1)\sqrt{I(\pi)}$, so, as $c_{k}>0$, $|c_{k}d_{k}|\le c_{k}(2R_{\mu}+1)\sqrt{I(\pi)}$; the series of these bounds converges with sum $c\,(2R_{\mu}+1)\sqrt{I(\pi)}$ and $K$-th partial sum $\sigma_{K}(2R_{\mu}+1)\sqrt{I(\pi)}$, by Elementary Properties of Series of Real Numbers §linearity. Step 2(T) in $E=\mathbb{R}$ gives

$$\bigl|\bigl(\psi(\nu)-\psi(\mu)\bigr)-\bigl(\psi_{K}(\nu)-\psi_{K}(\mu)\bigr)\bigr|\le(2R_{\mu}+1)\,r_{K}\sqrt{I(\pi)} .$$

Second, the field tail. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §linear, $\mathcal{J}(\eta,\pi)=\mathcal{J}(s_{K},\pi)+\mathcal{J}(\eta-s_{K},\pi)$, and by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §bound and Step 5(iii) with $M=R_{\mu}$ (admissible by Step 1(ii)),

$$\bigl|\mathcal{J}(\eta-s_{K},\pi)\bigr|\le\lVert\eta-s_{K}\rVert_{\mu}\sqrt{I(\pi)}\le2R_{\mu}\,r_{K}\sqrt{I(\pi)} .$$

Third, the head: $|\psi_{K}(\nu)-\psi_{K}(\mu)-\mathcal{J}(s_{K},\pi)|\le\varepsilon\cdot2^{-1}\sqrt{I(\pi)}$ by the choice of $\theta_{1}$, since $I(\pi)<\theta_{1}^{2}$. Writing $\psi(\nu)-\psi(\mu)-\mathcal{J}(\eta,\pi)$ as the sum of the head expression, the real tail difference and $-\mathcal{J}(\eta-s_{K},\pi)$, the triangle inequality for the absolute value gives

$$\bigl|\psi(\nu)-\psi(\mu)-\mathcal{J}(\eta,\pi)\bigr|\le\Bigl(\varepsilon\cdot2^{-1}+(4R_{\mu}+1)\,r_{K}\Bigr)\sqrt{I(\pi)}\le\varepsilon\sqrt{I(\pi)} .$$

Hence $\psi$ is differentiable along couplings at $\mu$ with gradient $\eta$, and $\nabla\psi(\mu)=\eta$ by the uniqueness in Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §gradient. By Step 5(iv), $\nabla\psi(\mu)\in T_{\mu}$, so property (b) holds. Since the optimal maps $S_{k}$ were arbitrary, this also proves clause 3: for any choice of optimal maps $S_{k}$ from $\mu$ to $\mu_{k}$, the series $\sum_{k}2c_{k}(\mathrm{id}-S_{k})$ converges in $L^{2}(\mu;\mathbb{R}^{d})$ (in the sense of series in that space) to $\nabla\psi(\mu)$ by Step 5(ii), and $\lVert\nabla\psi(\mu)\rVert_{\mu}\le2\sum_{k=1}^{\infty}c_{k}W_{2}(\mu,\mu_{k})$ by the same step.

Step 7 (Property (c)). Let $\mu\in Q$, let $(\rho_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$, and let $\pi_{n}\in\Pi(\rho_{n},\mu)$ with $(I(\pi_{n}))$ converging to $0$. For each $n$ fix the optimal maps $S^{\rho_{n}}_{k}$ of Step 5, and write $g_{n}=\nabla\psi(\rho_{n})=\eta_{\rho_{n}}$ and $g=\nabla\psi(\mu)=\eta_{\mu}$ (Step 6), and $D_{n}$ for the discrepancy of $g_{n}$ and $g$ along $\pi_{n}$, a nonnegative real number by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined. For $K\in\mathbb{N}$ write $h_{n,K}=s^{\rho_{n}}_{K}=\nabla\psi_{K}(\rho_{n})$ and $h_{K}=s^{\mu}_{K}=\nabla\psi_{K}(\mu)$ (Step 5(v)), and $D_{n,K}$ for the discrepancy of $h_{n,K}$ and $h_{K}$ along $\pi_{n}$.

Comparison. Fix $n$ and $K$. Choose representatives of $h_{n,K}$, $v=g_{n}-h_{n,K}$, $h_{K}$ and $w=g-h_{K}$; by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space, applied on the probability spaces $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\rho_{n})$ and $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, the pointwise sums $h_{n,K}+v$ and $h_{K}+w$ represent $g_{n}$ and $g$, and the discrepancies do not depend on the representatives (The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined). For every $z$, $g_{n}(x)-g(y)=\bigl(h_{n,K}(x)-h_{K}(y)\bigr)+v(x)+(-1)w(y)$, so by the triangle inequality and homogeneity of the Euclidean norm (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle, Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §homogeneity) and the real inequality $(p+q+t)^{2}\le3(p^{2}+q^{2}+t^{2})$,

$$\lVert g_{n}(x)-g(y)\rVert^{2}\le3\lVert h_{n,K}(x)-h_{K}(y)\rVert^{2}+3\lVert v(x)-0_{\mathbb{R}^{d}}\rVert^{2}+3\lVert0_{\mathbb{R}^{d}}-w(y)\rVert^{2}.$$

The three functions on the right are nonnegative and Borel (The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined, with the zero fields of $L^{2}(\mu;\mathbb{R}^{d})$ and of $L^{2}(\rho_{n};\mathbb{R}^{d})$, of norm $0$), so integrating against $\pi_{n}$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and bounding the last two discrepancies by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined,

$$D_{n}\le3D_{n,K}+6\lVert g_{n}-h_{n,K}\rVert_{\rho_{n}}^{2}+6\lVert g-h_{K}\rVert_{\mu}^{2}.$$

Tails. Since $(I(\pi_{n}))$ converges to $0$, there is $N_{0}\in\mathbb{N}$ with $I(\pi_{n})<1$ for $n\ge N_{0}$. For such $n$, $W_{2}(\mu,\rho_{n})=W_{2}(\rho_{n},\mu)\le\sqrt{I(\pi_{n})}\le1$ by Step 1(i) and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $W_{2}(\rho_{n},\mu_{k})\le R_{\mu}+1$ for every $k$ by Step 1(iii); also $W_{2}(\mu,\mu_{k})\le R_{\mu}+1$ by Step 1(ii). Step 5(iii) with $M=R_{\mu}+1$ at $\rho_{n}$ and at $\mu$ gives $\lVert g_{n}-h_{n,K}\rVert_{\rho_{n}}\le2(R_{\mu}+1)r_{K}$ and $\lVert g-h_{K}\rVert_{\mu}\le2(R_{\mu}+1)r_{K}$, whence by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field

$$D_{n}\le3D_{n,K}+48\,(R_{\mu}+1)^{2}\,r_{K}^{2}\qquad(n\ge N_{0},\ K\in\mathbb{N}).$$

Conclusion. Let $\varepsilon\in\mathbb{R}$ be positive. The order of choice is: first $K$, then $N_{1}$. Choose $K$ with $48(R_{\mu}+1)^{2}r_{K}^{2}<\varepsilon\cdot2^{-1}$ (opening paragraph). By Step 5(v), property (c) of the intrinsic test function $\psi_{K}$ on $Q$, applied to $\mu$, $(\rho_{n})$ and $(\pi_{n})$, shows that $(D_{n,K})_{n\in\mathbb{N}}$ converges to $0$, so there is $N_{1}$ with $3D_{n,K}<\varepsilon\cdot2^{-1}$ for $n\ge N_{1}$ (Arithmetic of Limits of Real Sequences §scalar). For $n\ge\max\{N_{0},N_{1}\}$ we get $0\le D_{n}<\varepsilon$. Hence $(D_{n})$ converges to $0$ and property (c) holds.

Step 8 (Properties (d) and (e), and clause 4). First a vector of weighted means. For $k\in\mathbb{N}$ and $i\in[d]$, $|m(\mu_{k})_{i}|\le\lVert m(\mu_{k})\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate, and $\lVert m(\mu_{k})\rVert^{2}\le M_{2}(\mu_{k})\le B$ by The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean, so $\lVert m(\mu_{k})\rVert\le\sqrt{B}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Thus $|c_{k}m(\mu_{k})_{i}|\le\sqrt{B}\,c_{k}$, and the series of the bounds converges by Elementary Properties of Series of Real Numbers §linearity; by Step 2(S) the series $\sum_{k}c_{k}m(\mu_{k})_{i}$ converges, and we let $p\in\mathbb{R}^{d}$ be the point with coordinates $p_{i}=\sum_{k=1}^{\infty}c_{k}m(\mu_{k})_{i}$. For $a\in\mathbb{R}^{d}$, the dot product being the finite sum of the products of coordinates (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$), $c_{k}\,(a\cdot m(\mu_{k}))=\sum_{i=1}^{d}a_{i}\,c_{k}m(\mu_{k})_{i}$, so, by Elementary Properties of Series of Real Numbers §linearity applied $d$ times (induction on the number of summands), the series $\sum_{k}c_{k}\,(a\cdot m(\mu_{k}))$ converges with sum $\sum_{i=1}^{d}a_{i}p_{i}=a\cdot p$.

Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$; the maps $\tau_{a}$ are the translations. The translation $\tau_{0_{\mathbb{R}^{d}}}$ is the identity map, so $(\tau_{0_{\mathbb{R}^{d}}})_{\#}\mu_{k}=\mu_{k}$ by the definition of push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, and The Squared Wasserstein Distance to a Fixed Measure: a One-Sided Bound Along Couplings, Differentiability at a Uniquely Mapped Source, and the Translation and Centring Identities §translation, read with $\rho$, $\mu_{k}$, $a$ and $b=0_{\mathbb{R}^{d}}$ in place of its $\mu$, $\nu$, $a$, $b$, gives, since $a-0_{\mathbb{R}^{d}}=a$ and by claims 4 and 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$,

$$W_{2}\bigl((\tau_{a})_{\#}\rho,\mu_{k}\bigr)^{2}=W_{2}(\rho,\mu_{k})^{2}+2\,\bigl(a\cdot m(\rho)\bigr)-2\,\bigl(a\cdot m(\mu_{k})\bigr)+\lVert a\rVert^{2}.$$

Multiplying by $c_{k}$ and summing, the four series $\sum_{k}c_{k}W_{2}(\rho,\mu_{k})^{2}$, $\sum_{k}c_{k}\cdot2(a\cdot m(\rho))$, $\sum_{k}(-2)\,c_{k}(a\cdot m(\mu_{k}))$ and $\sum_{k}c_{k}\lVert a\rVert^{2}$ converge with sums $\psi(\rho)$, $2c\,(a\cdot m(\rho))$, $-2\,(a\cdot p)$ and $c\lVert a\rVert^{2}$, by Step 3, the previous paragraph and Elementary Properties of Series of Real Numbers §linearity; by the same clause,

$$\phi_{\rho}(a)=\psi\bigl((\tau_{a})_{\#}\rho\bigr)=\psi(\rho)+2c\,\bigl(a\cdot m(\rho)\bigr)-2\,(a\cdot p)+c\,\lVert a\rVert^{2}.$$

Now $\lVert a\rVert^{2}=a\cdot a$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, $(2cI_{d})a=2c\,(I_{d}a)=2c\,a$ by claims 1 and 2 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, and $a\cdot(2c\,a)=2c\,(a\cdot a)$ by claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$; so $c\lVert a\rVert^{2}=\tfrac{1}{2}\,a\cdot\bigl((2cI_{d})a\bigr)$. With $q=2c\,m(\rho)-2p$, claims 1 and 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ give $2c\,(a\cdot m(\rho))-2\,(a\cdot p)=q\cdot a$. Hence

$$\phi_{\rho}(a)=\tfrac{1}{2}\,a\cdot\bigl((2cI_{d})a\bigr)+q\cdot a+\psi(\rho)\qquad(a\in\mathbb{R}^{d}).$$

The matrix $2cI_{d}$ is symmetric, its entries being unchanged when the indices are interchanged, so it lies in $\mathcal{S}(d)$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, read in dimension $d$ with $M=2cI_{d}$, the vector $q$ above and the constant $\psi(\rho)$ in place of its $c$, the function $\phi_{\rho}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $D^{2}\phi_{\rho}(a)=2cI_{d}$ for every $a$. So property (d) holds, and $D^{2}\phi_{\rho}(0_{\mathbb{R}^{d}})=2cI_{d}$ for every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

For property (e), let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\varepsilon$ be positive; put $\theta=\varepsilon$. For every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $D^{2}\phi_{\nu}(0_{\mathbb{R}^{d}})-D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})=2cI_{d}-2cI_{d}$ is the zero matrix $0_{d}$, entrywise by Difference of Real Matrices, and $\lVert0_{d}\rVert=0$ by claim 4 of Properties of the Norm of a Symmetric Real Matrix; so the distance in property (e) is $0<\varepsilon$.

Conclusion. Steps 4, 6, 7 and 8 verify properties (a) to (e) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test, so $\psi$ is an intrinsic test function on $Q$: this is clause 2. Clause 1 is Step 3 and clause 3 was proved in Step 6. Finally, for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian and Step 8 give $H_{\psi}(\mu)=D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})=2c\,I_{d}$, which is clause 4.