# rt.representation theory – Independence between \$X_{n-k}\$ and \$sumlimits_i Y_{n-i}-Y_{n-k}\$

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 $$sumlimits_i X_{n-i:n}-X_{n-k:n}$$ are independent.

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