Central Limit Theorem of the Smoothed Empirical Distribution Functions for Asymptotically Stationary Absolutely Regular Stochastic Processes

Let F^n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n. A central limit theorem is established for the target statistic F^n(ξ^n), where the underlying random vector forms an asymptotically stationary absolutely regular stochastic process, and ξ^n is an estimator of a multivariate parameter ξ by using a vector of U-statistics. The results obtained extend or generalize previous results from the stationary univariate case to the asymptotically stationary multivariate case. An example of asymptotically stationary absolutely regular multivariate ARMA process and an example of a useful estimation of F(ξ) are given in the applications