Asymptotic distributions of multivariate intermediate order statistics

Let $\{X_n=(X_n^{(1)},\ldots,X_n^{(d)}),n\ge 1\}$ be independent identically distributed random vectors with a common nondegenerate distribution function and for each $n\ge 1$ and each $k=1,\ldots,d$, denote $X_{1;n}^{(k)}\le\cdots\le X_{n;n}^{(k)}$ as the order statistics of $X_1^{(k)},\ldots,X_n^{(k)}$. Suppose that ranks $r_n=(r_n^{(1)},\ldots,r_n^{(d)})$ satisfy $r_n^{(k)} \to\infty$ nondecreasingly, $r_n^{(k)}/n\to 0$ and $r_n^{(k)}/\sum_{l=1}^d r_n^{(l)}\to m^{(k)}>0$ for each $k=1,\ldots,d$ and let $X_{r_n;n}= (X_{r_n^{(1)};n}^{(1)},\ldots,X_{r_n^{(d)};n}^{(d)})$. This paper is to find out the class of limiting distributions of $\{X_{r_n;n}\}$ after suitable normalizing and centering, and give necessary and sufficient conditions of weak convergence.