@IosifPinelis (i) & (iii) yes but there is a way to generally proof that regardless the distribution the independence proof will still hold, check lemma 3.2.3 in extreme value theory an introduction. (ii) the summation is from 0 to k. but regardless the same I need to see if there is dependence between the OS $X_{n-k:n}$ and the difference $Y_{n-i:n}-Y_{n-k:n}$ !

@DieterKadelka OS are done separately but there is dependence between the main variables. do you have a way to reason why this dependence is valid?!