# Difference of order statistics between independent samples.

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

\begin{align*} \mathbb{P}\left(X_{(k)} \mathbb{P}\left(X_{(k)}

• 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? – RRL Feb 5 at 13:44