Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 735436, 18 pages
Research Article

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

Echarif Elharfaoui1,2 and Michel Harel1

1Laboratoire de Statistique et Probabilités, CNRS, (UMR 5219), Université Paul Sabatier, Toulouse Cedex 931062, France
2IUFM du Limousin, 209 Boulevard de Vanteaux, Limoges Cedex 87036, France

Received 9 March 2007; Revised 27 September 2007; Accepted 24 October 2007

Academic Editor: Andrew Rosalsky

Copyright © 2008 Echarif Elharfaoui and Michel Harel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.