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

Abstract

Let $(X_j)^\infty_{j = 1}$ be a stationary, mean-zero Gaussian process with covariances $r(k) = EX_{k + 1} X_1$ satisfying $r(0) = 1$ and $r(k) = k^{-D}L(k)$ where $D$ is small and $L$ is slowly varying at infinity. Consider the two-parameter empirical process for $G(X_j),$ $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where $G$ is any measurable function. Noncentral limit theorems are obtained for $F_N(x, t)$ and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and $U$-statistics based on the $G(X_j)$'s. The limiting processes are structurally different from those encountered in the i.i.d. case.