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 $$\textbf{E}sup_{\mu \in S^{d-1}} W_2^2(\hat{X}\mu^T,X\mu^T) ?$$

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

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