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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 $A$.

Related

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