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

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, let $\mathcal{P}_{2}(X)$ be the set of probability measures on $X$ with finite second moment and $W_{2}$ the quadratic Wasserstein distance on it. Let $(\mu_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(X)$.

1. (Weak convergence)¶ If $\mu\in\mathcal{P}_{2}(X)$ and $\lim_{j}W_{2}(\mu_{j},\mu)=0$, then $\mu_{j}\Rightarrow\mu$.

2. (Quadratic growth)¶ Let $\mu\in\mathcal{P}_{2}(X)$ with $\lim_{j}W_{2}(\mu_{j},\mu)=0$, let $h:X\to\mathbb{R}$ be continuous, and let $A\in\mathbb{R}$ satisfy $|h(x)|\le A(1+|x|^{2})$ for every $x\in X$. Then $h$ is integrable with respect to $\mu$ and to every $\mu_{j}$, and $\lim_{j}\int_{X}h\,d\mu_{j}=\int_{X}h\,d\mu$.

3. (Convergence)¶ Let $\mu\in\mathcal{P}(X)$ with $\mu_{j}\Rightarrow\mu$, and suppose that for every real $\varepsilon>0$ there is a real $K>0$ with

$$\int_{\{x:\,K<|x|\}}|x|^{2}\,\mu_{j}(dx)<\varepsilon\qquad\text{for every }j\in\mathbb{N}.$$

Then $\mu\in\mathcal{P}_{2}(X)$ and $\lim_{j}W_{2}(\mu_{j},\mu)=0$.

4. (Compactness)¶ Suppose $(\mu_{j})_{j\in\mathbb{N}}$ is tight in $(X,d)$ and that for every real $\varepsilon>0$ there is a real $K>0$ with $\int_{\{x:\,K<|x|\}}|x|^{2}\,\mu_{j}(dx)<\varepsilon$ for every $j\in\mathbb{N}$. Then there are $\mu\in\mathcal{P}_{2}(X)$ and a strictly increasing sequence $(j_{i})_{i\in\mathbb{N}}$ in $\mathbb{N}$ with $\lim_{i}W_{2}(\mu_{j_{i}},\mu)=0$.