# Expectation of maximal Wasserstein distance between empirical distribution and a pdf

Let $$P$$ be a continuous probability distribution on $$R^d$$, $$X$$ the random variable $$\sim P$$, and $$\hat{X}$$ be n i.i.d samples drawn according to $$P$$. We have another variable $$\mu \in S^{d-1}$$. Do we have some result about $$\textbf{E}sup_{\mu \in S^{d-1}} W_2^2(\hat{X}\mu^T,X\mu^T) ?$$

• What is the sup over? – Aryeh Kontorovich Jan 26 at 18:57
• But these are random?.. – Aryeh Kontorovich Jan 26 at 19:32
• I made a mistake about the $sup$, now I give the original problem. – Will Cai Jan 28 at 1:37