Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $m$ be a dimension, with the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{m}),W_{2})$ and the set $\mathcal{P}(\mathbb{R}^{m})$ of Borel probability measures fixed there. Weak convergence $\mu_{n}\Rightarrow\mu$ is that of that definition on $\mathbb{R}^{m}$ with the Euclidean distance, limits of real sequences are those of that definition, integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and Borel as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.

1. (Convergence)¶ Let $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{m})$ for $n\in\mathbb{N}$ and $\mu\in\mathcal{P}(\mathbb{R}^{m})$ satisfy $\mu_{n}\Rightarrow\mu$, and suppose that for every positive $\varepsilon\in\mathbb{R}$ there is a positive $K\in\mathbb{R}$ with

Then $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$ and the sequence $(W_{2}(\mu_{n},\mu))_{n\in\mathbb{N}}$ has limit $0$.

2. (Compactness)¶ Let $h:\mathbb{R}^{m}\to\mathbb{R}$ be Borel and bounded below, and superquadratic: for every positive $M\in\mathbb{R}$ there is a positive $K\in\mathbb{R}$ with $M\lVert x\rVert^{2}\le h(x)$ for every $x\in\mathbb{R}^{m}$ with $K\le\lVert x\rVert$. Let $c\in\mathbb{R}$, and let $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{m})$ for $n\in\mathbb{N}$ be such that $h$ is $\mu_{n}$-integrable with $\int h\,d\mu_{n}\le c$ for every $n$. Then there are $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$ and a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ such that $(W_{2}(\mu_{n_{k}},\mu))_{k\in\mathbb{N}}$ has limit $0$.