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Let (Xⱼ) ^₉ = ₁ be a stationary, mean-zero Gaussian process with covariances r (k) = EX₊ + 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), \FN (x, t) = 1{N ^ Nt ₉ = ₁ 1\G (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.
Dehling et al. (Fri,) studied this question.