Asymptotic distributions of multivariate intermediate order statistics

Let {X-n = (X-n((1)),...,X-n((d))), n greater than or equal to 1} be independent identically distributed random vectors with a common nondegenerate distribution function and for each n greater than or equal to 1 and each k = 1,...,d, denote X-1;n((k)) less than or equal to...less than or equal to X-n;n((k)) as the order statistics of X-1((k)),...,X-n((k)). Suppose that ranges r(n) = (r(n)((1)),...,r(n)((d))) satisfy r(n)((k)) --> infinity nondecreasingly, r(n)((k))/n --> 0 and r(n)((k))/Sigma(l=1)(d)r(n)((l)) --> m((k)) > 0 for each k = 1,...,d and let X-rn;n = (X-rn(1;n)((1)),...,X-rn(d;n)((d))). This paper is to find out the class of limiting distributions of {X-rn;n} after suitable normalizing and centering, and give necessary and sufficient conditions of weak convergence.