0
$\begingroup$

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i.i.d from $\mu$. Consider the empirical measure $\hat{\mu}_N := (1/N)\sum_{i=1}^N\delta_{x_i}$ thus obtained.

Question

Given $\epsilon > 0$, is it possible to construct a subset $B$ of $X$ such that $$W_1(\hat{\mu}_N,\mu|_B) \le \epsilon? $$

N.B.: Here, $\mu|_B: A \mapsto \dfrac{\mu(A \cap B)}{\mu(B)}$ defines the restriction of $\mu$ on $B$, and $W_1$ is the Wassertstein distance.

Observation

On second though, it seems the $\epsilon$-blowup of $S_N:=\{x_1,\ldots,x_N\}$ defined by $S_N^\epsilon := \{x \in X | \min_{i=1,\ldots,N} d(x,x_i) \le \epsilon\} = \cup_{i=1}^N \operatorname{Ball}_X(x_i;\epsilon)$ might do the trick.

Question 2

Can Every distribution $\nu$ on $X$ with $W_1(\hat{\mu}_N,\nu) \le \epsilon$ be realized in such a way (i.e by constructing some appropriate $B \subseteq X$ and setting $\nu = \mu|_B$) ?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.