Hájek-Sidák approach to the asymptotic distribution of multivariate rank order statistics

Let X[alpha] = (X1[alpha],..., Xp[alpha]), 1 <= [alpha] <= N[nu], [nu] >= 1 be N[nu] independent observations from a density function f(x) where x [set membership, variant] Rp, the p-dimensional real space. Let R[nu]j[alpha] denote the rank of Xj[alpha] in the ordered array of Xj1 ,..., XjN[nu]; 1 <= j <= p and consider the multivariate rank order statistics where the constants, c[nu][alpha], 1 <= [alpha] <= N[nu] satisfy the Noether condition and the scores, a[nu]j([alpha]), 1 <= j <= p, 1 <= [alpha] <= N[nu] converge as [nu] --> [infinity], for each j, in quadratic mean to a nonconstant, square integrable function [pi]j(u), 0 < u < 1. Under the hypothesis of randomness, the joint asymptotic conditional and uncoditional normality of the statistics T[nu]j, 1 <= j <= p is established. Further, under mild conditions on the underlying density functions and assuming contiguous location shift alternatives, the joint asymptotic normality of these statistics is also established.