Law of Large Numbers for Empirical Measures in the Wasserstein Distance

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 $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$, the set of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. For $N\in\mathbb{N}$, $\nu^{\otimes N}\in\mathcal{P}(\mathbb{R}^{qN})$ is the tensor power, $\mu^{N}_{x}$ is the empirical measure of $x\in\mathbb{R}^{qN}$, and $W_{2}$ is the Wasserstein distance.

1. (Root-mean-square distance)¶ For every $N\in\mathbb{N}$ the function $x\mapsto W_{2}(\mu^{N}_{x},\nu)^{2}$ on $\mathbb{R}^{qN}$, Borel by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §distance and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, is integrable with respect to $\nu^{\otimes N}$. We write $\phi_{N}(\nu)$ for the nonnegative square root of its integral,

2. (Law of large numbers)¶ The sequence $(\phi_{N}(\nu))_{N\in\mathbb{N}}$ converges to $0$.