Two-Sample U-Statistic Processes for Long-Range Dependent Data

Motivated by some common-change point tests, we investigate the asymptotic distribution of the U-statistic process U_n(t)=∑_i=1^[nt]∑_j=[nt]+1^n h(X_i,X_j), 0≤ t≤ 1, when the underlying data are long-range dependent. We present two approaches, one based on an expansion of the kernel h(x,y) into Hermite polynomials, the other based on an empirical process representation of the U-statistic. Together, the two approaches cover a wide range of kernels, including all kernels commonly used in applications.