An invariance principle for processes indexed by two parameters and some statistical applications

Let D((0 l] ² ) denote the space of all functions on (0,l] ² with no discontinuities of the second kind. We prove weak invariance principles in the space D((0,l] ² ) for processes of the form ∫h(H _n+m (t))dF _n (t), m, n ≤ 1, where F _n and G _m are two independent empirical distribution functions of independent, identically distributed sequences of random variables, H _n+m = (n+m+I) ^-1 (nF _n + mG _m ), and where h belongs to a certain class of functions on the open unit interval. The appropriate topology in D(((0,l] ² ) is given by uniform convergence on compact sets. This type of processes is central in nonparametric statistics having applications to two-sample linear rank statistics and signed rank statistics.