Variance of the Empirical Mass of a Borel Set under a Tensor Power

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 $q,N\in\mathbb{N}$, let $\rho\in\mathcal{P}(\mathbb{R}^{q})$ with tensor power $\rho^{\otimes N}\in\mathcal{P}(\mathbb{R}^{qN})$, let $B\in\mathcal{B}(\mathbb{R}^{q})$ and $c=\rho(B)$, and let $\mu^{N}_{x}$ be the empirical measure of $x\in\mathbb{R}^{qN}$. Then¶ the function $x\mapsto\mu^{N}_{x}(B)$ on $\mathbb{R}^{qN}$ is Borel with values in $[0,1]$, and the following hold.

1. (Mean-square deviation)¶

2. (Mean absolute deviation)¶