Self-normalized central limit theorem and estimation of variance of partial sums for negative dependent random variables

  • Zhang Lixin 
  • Shi Strongway 
Article

Abstract

Let {Xn, n≥1} be a stationary LNQD or NA sequence satisfying EX1 = μ, EX12<∞ and (Var Sn)/n→σ2 as n→∞. In this paper a class of self-normalized central limit theorems and estimators of Var Sn are studied. The weak and strong consistency of the estimators of Var Sn are presented.

MR Subject Classification

60F05 60F15 

Keywords

estimation negative dependence consistency variance of partial sums 

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References

  1. 1.
    Joag-Dev, K., Proschan, F., Negative association of random variables, with application, Annals Statistics, 1983,11:286–295.MathSciNetGoogle Scholar
  2. 2.
    Lin, Z. Y., An invariance principle for NA random variables, Chinese Sci. Bull., 1997,42:359–364.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Matula, P., A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett., 1992,15:209–213.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Móricz, F., A general moment inequality for the maximum of partial sums of single series, Acta. Sci. Math., 1982,44:67–75.MATHGoogle Scholar
  5. 5.
    Newman, C. M., Wright, A. L., An invariance principle for certain dependent sequences, Annals Statistics, 1981,9:671–675.MATHMathSciNetGoogle Scholar
  6. 6.
    Peligrad, M., Shao, Q. M., Estimation of the variance of partial sums for ρ-mixing random variables, J. Multivariate Anal., 1995,52:140–157.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Peligrad, M., Suresh, R., Estimation of variance of partial sums of an associated sequence of random variables, Stochastic process. Appl., 1995,56:307–319.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Shao, Q. M., A comparison theorem on maximal inequalities between negatively associated and independent random variables, Stochastic Process. Appl. 1999,86:139–148.CrossRefGoogle Scholar
  9. 9.
    Su, C., Zhao, L. C., Wang, Y. B., Inequalities and weak convergence for negatively associated sequence, Science of China, Ser. A, 1997,40:172–182.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhang, L. X., A functional central limit theorem for asymptotically negatively dependent random fields, Acta Math. Hungar., 2000,86:237–259.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Zhang, L. X., The weak convergence for functions of negatively associated random variables, J. Multivariate Anal., 2001,78:272–298.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities 2002

Authors and Affiliations

  • Zhang Lixin 
    • 1
  • Shi Strongway 
    • 1
  1. 1.Dept. of Math.Zhejiang Univ., Xixi CampusHangzhouChina

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