The Law of Large Numbers for Empirical Measures is Uniform on Wasserstein-Compact Sets

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used, let $q\in\mathbb{N}$, and for $N\in\mathbb{N}$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ let $\phi_{N}(\nu)$ be the root-mean-square distance of Law of Large Numbers for Empirical Measures in the Wasserstein Distance §finite; $W_{2}$ is the Wasserstein distance.

1. (Lipschitz dependence on the measure)¶ For every $N\in\mathbb{N}$ and all $\nu,\nu'\in\mathcal{P}_{2}(\mathbb{R}^{q})$, $|\phi_{N}(\nu)-\phi_{N}(\nu')|\le2\,W_{2}(\nu,\nu')$.

2. (Uniformity on compact sets)¶ Let $K$ be a compact subset of the metric space $(\mathcal{P}_{2}(\mathbb{R}^{q}),W_{2})$ and let $\varepsilon\in\mathbb{R}$ be positive. Then there is $N_{0}\in\mathbb{N}$ such that $\phi_{N}(\nu)\le\varepsilon$ for every $N\in\mathbb{N}$ with $N_{0}\le N$ and every $\nu\in K$.