The Empirical Process of some Long-Range Dependent Sequences with an Application to U-Statistics

Let (Xⱼ)^∞j = 1 be a stationary, mean-zero Gaussian process with covariances r(k) = EXk + 1 X₁ satisfying $r(0) = 1$ and r(k) = k-DL(k) where D is small and L is slowly varying at infinity. Consider the two-parameter empirical process for G(Xⱼ), (x, t) = 1/N ∑Ntj = 1 1(Xⱼ) ≤ x\ - P(G(X₁) ≤ x) ; // -∞ < x < + ∞, 0 ≤ t ≤ 1\, where G is any measurable function. Noncentral limit theorems are obtained for FN(x, t) and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and U-statistics based on the G(Xⱼ)'s. The limiting processes are structurally different from those encountered in the i.i.d. case.