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Let $F$ be some cumulative distribution function. Suppose we draw independently two i.i.d. $n$-samples with distribution $F$, say $(X_k)_{1\leq k \leq n}$, $(Y_k)_{1\leq k \leq n}$. Denote $X_{(i)}$, $Y_{(i)}$ their $i$-th order statistics respectively. Is it true that the random variables \begin{align*} Z_i = X_{(i)} - Y_{(i)} \quad \text{$i = 1,\dots,n$} \end{align*} are independent?

EDIT: This question emerges when one wants to derive converge properties of the empirical Wasserstein distance between two independent but identically distributed samples.

The statement is false. For $k<l$, $\mathbb{P}(X_{(k)}<X_{(l)})=1$ whereas $\mathbb{P}(Y_{(k)}>X_{(l)})>0$. One concludes by noticing that, conditioning on $\{X_{(l)}<Y_{(l)}\}$, it is more likely that $X_{(k)}<Y_{(k)}$, i.e.:

\begin{align*} \mathbb{P}\left(X_{(k)}<Y_{(k)} | X_{(l)}<Y_{(l)}\right) > \mathbb{P}\left(X_{(k)}<Y_{(k)}\right) = 1/2 \end{align*}

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  • $\begingroup$ Try adding some context or example of personal effort to avoid closure. For example what is the joint PDF just for order statistics $X_{(i)}$ and $X_{(j)}$? Are these even independent? $\endgroup$ – RRL Feb 5 at 13:44

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