Open Access

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

Journal of Inequalities and Applications20062006:58317

DOI: 10.1155/JIA/2006/58317

Received: 21 April 2005

Accepted: 11 December 2005

Published: 25 May 2006

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F58317/MediaObjects/13660_2005_Article_1614_IEq1_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F58317/MediaObjects/13660_2005_Article_1614_IEq2_HTML.gif 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]

Authors’ Affiliations

(1)
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute
(2)
Department of Mathematics and Statistics, University of Hyderabad

References

  1. Barlow RE, Proschan F: Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York; 1981.MATHGoogle 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)90080-6MATHMathSciNetView ArticleGoogle Scholar
  3. Chung KL: A Course in Probability Theory. Academic Press, New York; 1974:xii+365.MATHGoogle 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/1176993303MATHMathSciNetView ArticleGoogle 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)00226-3MATHMathSciNetView ArticleGoogle 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/1177698701MATHMathSciNetView ArticleGoogle Scholar
  7. Louhichi S: Convergence rates in the strong law for associated random variables. Probability and Mathematical Statistics 2000,20(1):203–214.MATHMathSciNetGoogle 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:1011315404181MATHMathSciNetView ArticleGoogle Scholar
  9. Newman CM: Normal fluctuations and the FKG inequalities. Communications in Mathematical Physics 1980,74(2):119–128. 10.1007/BF01197754MATHMathSciNetView ArticleGoogle Scholar
  10. Newman CM: A general central limit theorem for FKG systems. Communications in Mathematical Physics 1983,91(1):75–80. 10.1007/BF01206051MATHMathSciNetView ArticleGoogle 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.View ArticleGoogle 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/1176994374MATHMathSciNetView ArticleGoogle 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)00025-1MATHMathSciNetView ArticleGoogle 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

Copyright

© I. Dewan and B. L. S. P. Rao 2006

This article is published under license to BioMed Central Ltd. 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.