# Mean of the product of order statistics.

Let $$X,Y$$ be two real valued random variables with first and second moments. Let $$(X_1,\dots,X_n)$$ i.i.d $$X$$ and $$(Y_1,\dots,Y_n)$$ i.i.d $$Y$$ be to independent $$n$$-samples. Denote $$\left(X_{(1)},\dots,X_{(n)}\right)$$, $$\left(Y_{(1)},\dots,Y_{(n)}\right)$$ their order statistics. How can one prove that: \begin{align*} \frac{1}{n}\sum_{i=1}^nX_{(i)}Y_{(i)} = \mu_n^X\mu_n^Y+\sigma_n^X\sigma_n^Y + o_{P}\left(\frac{1}{n}\right) \end{align*} where $$\mu_n^X$$ and $$\sigma_n^X$$ (resp $$\mu_n^Y$$ and $$\sigma_n^Y$$) are the empirical mean and standard deviation of $$(X_1,\dots,X_n)$$ (resp $$(Y_1,\dots,Y_n)$$).