Asymptotic independence of components of multivariate extreme order statistics

Let $X_{ik}$ be the $(n-k+1)$-th order statistic for a sample of the $i$-th coordinates of independent random vectors $X^{(1)},\dots,X^{(n)}$ in $R^N$. We investigate necessary and sufficient conditions for asymptotic independence of $X_{ik_i}$, $i=1,\dots,N$, as $n\to\infty$ where $k_1,\dots,k_N$ are arbitrary fixed numbers.