# Empirical estimation of $\inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega)$, given i.i.d samples from $\mu$ and $\nu$

Let $$\mathcal X$$ be a Polish space and $$\Omega \subseteq \mathcal X^2$$ be open. Let $$\mu$$ and $$\nu$$ be probability measures, and consider the quantity $$c_\Omega(\mu,\nu)$$ defined by

$$c_\Omega(\mu,\nu) := \inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega).$$ This is nothing but the transportation distance between $$\mu$$ and $$\nu$$, for the binary ground cost given by $$c_\Omega(x,x') = 1$$ if $$(x,x') \in \Omega$$; $$c_\Omega(x,x') = 0$$ else. One can show (Strassen's theorem) that $$c_\Omega(\mu,\nu) = \sup_{\text{closes }A\subseteq \mathcal X}\mu(A)-\nu(A_\Omega),$$ where $$A_\Omega := \{x \in \mathcal X \mid \exists x' \in A,\;(x,x') \not\in \Omega\}$$. In particular, if $$\mathcal X^2\setminus\Omega = \{(x,x) \mid x \in \mathcal X\}$$ (i.e the "diagonal" of $$\mathcal X^2$$), then$$A_\Omega=A$$ and $$c_\Omega(\mu,\nu)$$ is nothing but the total-variation distance between $$\mu$$ and $$\nu$$.

# Question

Let $$\hat{\mu}_n$$ and $$\hat{\nu}_n$$ be empirical versions of $$\mu$$ and $$\nu$$, each based on an i.i.d sample of size $$n$$.

• When (and in what sense) does $$c_\Omega(\hat{\mu}_n,\hat{\nu}_n)$$ converge to $$c_\Omega(\mu,\nu)$$ and at what rate ?

• Same question when $$\mathcal X$$ is Banach, and $$\Omega = \{(x,x') \in \mathcal X^2 \mid \|x-x'\| > \alpha\}$$ for some $$\alpha \ge 0$$. Note that in this case, $$A_\Omega = A^\alpha := \{x \in \mathcal X \mid \exists x' \in A,\text{ with }\|x-x'\| \le \alpha\}$$, the $$\alpha$$-fattening of $$A$$.