By means of a general weak convergence theorem some invariance principles are proven for the multivariate sequential empirical process and for the multivariate rank order process w.r.t. stronger metrics than the generalized Skorohod metric. The underlying random variables are neither assumed to be independent nor to be stationary. These results are then applied to derive convergence of the weighted empirical cumulatives and for the weighted rank order process. Finally by a new representation asymptotic normality is proven for a general class of linear multivariate rank order statistics.
"Asymptotic Distributions of Multivariate Rank Order Statistics." Ann. Statist. 4 (5) 912 - 923, September, 1976. https://doi.org/10.1214/aos/1176343588