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Science in China Series A: Mathematics

, Volume 47, Issue 6, pp 821–830 | Cite as

A local probability exponential inequality for the large deviation of an empirical process indexed by an unbounded class of functions and its application

  • Dixin Zhang
Article
  • 46 Downloads

Abstract

A local probability exponential inequality for the tail of large deviation of an empirical process over an unbounded class of functions is proposed and studied. A new method of truncating the original probability space and a new symmetrization method are given. Using these methods, the local probability exponential inequalities for the tails of large deviations of empirical processes with non-i.i.d. independent samples over unbounded class of functions are established. Some applications of the inequalities are discussed. As an additional result of this paper, under the conditions of Kolmogorov theorem, the strong convergence results of Kolmogorov on sums of non-i.i.d. independent random variables are extended to the cases of empirical processes indexed by unbounded classes of functions, the local probability exponential inequalities and the laws of the logarithm for the empirical processes are obtained.

Keywords

empirical process large deviation unbounded class of functions local probability exponential inequality 

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References

  1. 1.
    Shorack, G. R., Wellner, J. A., Empirical Processes with Applications to Statistics, New York: Wiley, 1986.Google Scholar
  2. 2.
    Vaart, A. W., Wellner, J. A., Weak Convergence and Empirical Processes, New York: Springer-Verlag, 1996.MATHGoogle Scholar
  3. 3.
    Chen Xiru, An Introduction of Mathematical Statistics, Beijing: Science Press, 1997.Google Scholar
  4. 4.
    Giné, E., Zinn, J., Bootstrapping empirical processes, Ann. Probab., 1990, 18: 851–869.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Zhang, D. X., Li G., Bootstrap approximation for generalized U-processes, Chinese Science Bulletin, 1994, 39:1761–1765.MATHGoogle Scholar
  6. 6.
    Zhang, D. X., Cheng, P., Random weighting approximation of empirical processes and their applications, Chinese Science Bulletin, 1996, 41: 353–357.MATHMathSciNetGoogle Scholar
  7. 7.
    Zhang, D. X., Bayesian bootstraps for U-processes, hypothesis tests and convergence of Dirichlet U-processes, Statistica Sinica, 2001, 11: 463–478.MATHMathSciNetGoogle Scholar
  8. 8.
    Yuen, K. C., Zhu, L, Zhang, D. X., Comparing k cumulative incidence functions through resampling methods, Lifetime Data Analysis, 2002, 8: 401–412.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pollard, D., Convergence of Stochastic Processes, New York: Springer-Verlag, 1984.MATHGoogle Scholar
  10. 10.
    Talagrand, M., Sharper bounds for Gaussian and empirical processes, Ann. Probab., 1994, 22: 28–76.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Alexander, K. S., Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Probab., 1984, 12: 1041–1067.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Alexander, K. S., Correction: Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Probab., 1987, 15: 428–430.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Talagrand, M., New concentration inequalities on product spaces, Invent. Math., 1996, 126: 505–563.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bennett, G., Probability inequalities for the sum of independent random variables, Journal of the American Statistical Association, 1962, 57: 33–45.MATHCrossRefGoogle Scholar
  15. 15.
    Hoeffding, W., Probability inequalities for sums of bounded random variables, JASA., 1963, 58: 13–30.MATHMathSciNetGoogle Scholar
  16. 16.
    Devroye, L., Bounds for the uniform deviation of empirical measures, J. Multivariate Anal., 1982, 12: 72–79.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Dudley, R. M., Central limit theorems for empirical measures, Ann. Probab., 1978, 6: 899–929.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dudley, R. M., A Course on Empirical Processes, Lecture Notes in Mathematics 1097, New York: Springer-Verlag, 1984.Google Scholar
  19. 19.
    Dudley, R. M., Philipp, W., Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrsch. verw. Gebiete, 1983, 62, (6): 509–552.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ginć, E., Zinn, J., Some limit theorems for empirical processes, Ann.Probab., 1984, 12: 929–989.CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wu, L. M., Large deviations moderate deviations and LIL for empirical processes, Ann. Probab., 1994, 22:17–27.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities, Sem. Probab. XXXIII, LNM 1709, 1999, 120–219.Google Scholar
  23. 23.
    Alexander, K. S., Talagrand, M., The law of the iterated logarithm for empirical processes on Vapnik-Červonenkis classes, J. Multivariate Anal. 1989, 30: 155–166.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Chow, Y. S., Teicher, H., Probability Theory, New York: Springer-Verlag, 1988.MATHGoogle Scholar
  25. 25.
    Petrov, V. V., Sums of Independent Random Variables, New York: Springer-Verlag, 1975.Google Scholar
  26. 26.
    Stout, W., Almost Sure Convergence, New York: Academic Press, Inc, 1974MATHGoogle Scholar

Copyright information

© Science in China Press 2004

Authors and Affiliations

  1. 1.Department of FinanceBusiness School, Nanjing UniversityNanjingChina

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