Consider a triangular array Xn1, ..., Xnn, n[set membership, variant] 1, of rowwise independent random elements with values in a measurable space. Suppose there exists [theta] [set membership, variant] [0, 1)such that Xn1, ..., Xn[n[theta] ] have distribution [nu]1 and Xn[n[theta]] + 1(n), ..., Xnn have distribution [nu]2. Csörgo and Horváth derived an invariance principle for a one-time parameter process, which is the foundation of a test for H0: [theta] = 0 versus H1: [theta] [set membership, variant](0, 1). We are interested in the more complex test problem H0: [theta] [set membership, variant] [Theta]0 versus H1,: [theta] [set membership, variant] [Theta]0, where [Theta]0[subset, double equals](0, 1). To treat this new situation, we extend the Csörgo-Horváth result in proving a functional limit theorem for a suitable two-time parameter process. We briefly sketch several applications of our result. Especially, the power of the Csörgo-Horváth test is investigated in detail.
