If $(X_i,Y_i), i=1,\ldots,n,$ is i.i.d sample from the joint distribution $F$ and there is dependence between the two variables say $R$. Denote the order statistics for the two variables $X_{1:n},\ldots, X_{n:n}$ and $Y_{1:n},\ldots, Y_{n:n}$ respectively.

Now by Renyi's representation, it is possible to show that $X_{n-k:n}$ and $\sum\limits_i X_{n-i:n}-X_{n-k:n}$ are independent.

I want to check if there is independence or not between $X_{n-k:n}$ and $\sum\limits_i Y_{n-i:n}-Y_{n-k:n}$?

If $(X_i,Y_i), i=1,\ldots,n,$ is i.i.d sample from the joint distribution $F$ and there is dependence between the two variables say $R$. Denote the order statistics for the two variables $X_{1:n},\ldots, X_{n:n}$ and $Y_{1:n},\ldots, Y_{n:n}$ respectively.

Now by Renyi's representation, it is possible to show that $X_{n-k:n}$ and $\sum\limits_{i=0} ^{k} X_{n-i:n}-X_{n-k:n}$ are independent, where $k\in{1,\ldots,n-1}$.

I want to check if there is independence or not between $X_{n-k:n}$ and $\sum\limits_{i=0} ^{k} Y_{n-i:n}-Y_{n-k:n}$?