Skip to main content
Journal of Inequalities and Applications
Download PDF

Hajek-Renyi-type inequality for some nonmonotonic functions of associated random variables

Journal of Inequalities and Applications volume 2006, Article number: 58317 (2006) Cite this article

Abstract

Let be a sequence of nonmonotonic functions of associated random variables. We derive a Newman and Wright (1981) type of inequality for the maximum of partial sums of the sequence and a Hajek-Renyi-type inequality for nonmonotonic functions of associated random variables under some conditions. As an application, a strong law of large numbers is obtained for nonmonotonic functions of associated random varaibles.

[1234567891011121314]

References

  1. Barlow RE, Proschan F: Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York; 1981.

    MATH  Google Scholar 

  2. Birkel T: A note on the strong law of large numbers for positively dependent random variables. Statistics & Probability Letters 1988,7(1):17–20. 10.1016/0167-7152(88)90080-6

    Article  MATH  MathSciNet  Google Scholar 

  3. Chung KL: A Course in Probability Theory. Academic Press, New York; 1974:xii+365.

    MATH  Google Scholar 

  4. Cox JT, Grimmett G: Central limit theorems for associated random variables and the percolation model. The Annals of Probability 1984,12(2):514–528. 10.1214/aop/1176993303

    Article  MATH  MathSciNet  Google Scholar 

  5. Dewan I, Prakasa Rao BLS: Asymptotic normality for-statistics of associated random variables. Journal of Statistical Planning and Inference 2001,97(2):201–225. 10.1016/S0378-3758(00)00226-3

    Article  MATH  MathSciNet  Google Scholar 

  6. Esary J, Proschan F, Walkup D: Association of random variables, with applications. Annals of Mathematical Statistics 1967, 38: 1466–1474. 10.1214/aoms/1177698701

    Article  MATH  MathSciNet  Google Scholar 

  7. Louhichi S: Convergence rates in the strong law for associated random variables. Probability and Mathematical Statistics 2000,20(1):203–214.

    MATH  MathSciNet  Google Scholar 

  8. Matula P: Limit theorems for sums of nonmonotonic functions of associated random variables. Journal of Mathematical Sciences 2001,105(6):2590–2593. 10.1023/A:1011315404181

    Article  MATH  MathSciNet  Google Scholar 

  9. Newman CM: Normal fluctuations and the FKG inequalities. Communications in Mathematical Physics 1980,74(2):119–128. 10.1007/BF01197754

    Article  MATH  MathSciNet  Google Scholar 

  10. Newman CM: A general central limit theorem for FKG systems. Communications in Mathematical Physics 1983,91(1):75–80. 10.1007/BF01206051

    Article  MATH  MathSciNet  Google Scholar 

  11. Newman CM: Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Inequalities in Statistics and Probability (Lincoln, Neb, 1982). Volume 5. Edited by: Tong YL. Institute of Mathematical Statistics, California; 1984:127–140.

    Chapter  Google Scholar 

  12. Newman CM, Wright AL: An invariance principle for certain dependent sequences. The Annals of Probability 1981,9(4):671–675. 10.1214/aop/1176994374

    Article  MATH  MathSciNet  Google Scholar 

  13. Prakasa Rao BLS: Hajek-Renyi-type inequality for associated sequences. Statistics & Probability Letters 2002,57(2):139–143. 10.1016/S0167-7152(02)00025-1

    Article  MATH  MathSciNet  Google Scholar 

  14. Roussas GG: Positive and negative dependence with some statistical applications. In Asymptotics, Nonparametrics, and Time Series. Volume 158. Edited by: Ghosh S. Marcel Dekker, New York; 1999:757–788.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, New Delhi, 110016, India

    Isha Dewan

  2. Department of Mathematics and Statistics, University of Hyderabad, Hyderabad, 500046, India

    B. L. S. Prakasa Rao

Authors
  1. Isha Dewan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. L. S. Prakasa Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isha Dewan.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Dewan, I., Rao, B.L.S.P. Hajek-Renyi-type inequality for some nonmonotonic functions of associated random variables. J Inequal Appl 2006, 58317 (2006). https://doi.org/10.1155/JIA/2006/58317

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/JIA/2006/58317

Keywords