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Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 735436, 18 pages
http://dx.doi.org/10.1155/2008/735436
Research Article

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

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.

Linked References

  1. M. L. Puri and S. S. Ralescu, “Central limit theorem for perturbed empirical distribution functions evaluated at a random point,” Journal of Multivariate Analysis, vol. 19, no. 2, pp. 273–279, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. Sun, “Asymptotic behavior of the perturbed empirical distribution functions evaluated at a random point for absolutely regular sequences,” Journal of Multivariate Analysis, vol. 47, no. 2, pp. 230–249, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Sun, “Perturbed empirical distribution functions and quantiles under dependence,” Journal of Theoretical Probability, vol. 8, no. 4, pp. 763–777, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Harel and M. L. Puri, “Limiting behavior of U-statistics, V-statistics, and one sample rank order statistics for nonstationary absolutely regular processes,” Journal of Multivariate Analysis, vol. 30, no. 2, pp. 181–204, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Harel and M. L. Puri, “Weak invariance of generalized U-statistics for nonstationary absolutely regular processes,” Stochastic Processes and Their Applications, vol. 34, no. 2, pp. 341–360, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. R. Ducharme and M. Mint El Mouvid, “Almost sure convergence of the local linear estimator of the conditional cumulative distribution function,” Comptes Rendus de l'Académie des Sciences, vol. 333, no. 9, pp. 873–876, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. Oodaira and K.-I. Yoshihara, “The law of the iterated logarithm for stationary processes satisfying mixing conditions,” Kōdai Mathematical Seminar Reports, vol. 23, pp. 311–334, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Harel and M. L. Puri, “Nonparametric density estimators based on nonstationary absolutely regular random sequences,” Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 3, pp. 233–254, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Sun and C.-Y. Chiang, “Limiting behaviour of the perturbed empirical distribution functions evaluated at U-statistics for strongly mixing sequences of random variables,” Journal of Applied Mathematics and Stochastic Analysis, vol. 10, no. 1, pp. 3–20, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968. View at Zentralblatt MATH · View at MathSciNet
  11. K.-I. Yoshihara, “Limiting behavior of U-statistics for stationary, absolutely regular processes,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 35, no. 3, pp. 237–252, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Denker and G. Keller, “On U-statistics and V. Mises' statistics for weakly dependent processes,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 64, no. 4, pp. 505–522, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. Hoeffding, “A class of statistics with asymptotically normal distribution,” Annals of Mathematical Statistics, vol. 19, pp. 293–325, 1948. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Harel and M. L. Puri, “Conditional empirical processes defined by nonstationary absolutely regular sequences,” Journal of Multivariate Analysis, vol. 70, no. 2, pp. 250–285, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet