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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 $(Y_1,\dots,Y_n)$).

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