A Two-Dimensional Functional Permutational Central Limit Theorem for Linear Rank Statistics

Abstract

Some two-dimensional time-parameter stochastic processes are constructed from a sequence of linear rank statistics by considering their projections on the spaces generated by the (double) sequence of anti-ranks. Under appropriate regularity conditions, it is shown that these processes weakly converge to Brownian sheets in the Skorokhod $J_1$-topology on the $D^2\lbrack 0, 1 \rbrack$ space. This unifies and extends earlier one-dimensional invariance principles for linear rank statistics to the two-dimensional case. The case of contiguous alternatives is treated briefly.