Asymptotic Behavior of the Perturbed Empirical Distribution-Functions Evaluated at a Random Point for Absolutely Regular Sequences

Let F̂n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from smooth distribution function F. A central limit theorem is established for the target statistic F̂n(Un) where the underlying random variable form an absolutely regular stationary process and where {Un} is a sequence of U-statistics. The result obtained generalizes Puri and Ralescu (1986, J. Multivariate Anal.19, 273-279) under the iid set-up.