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Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

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  • Guodong Xing

    (Hunan University of Science and Engineering)

  • Shanchao Yang

    (Guangxi Normal University)

Abstract

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n −1/2 β n log 3/2 n in which β n can be particularly taken as (log log n)1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n −1/2(log log n)1/2log 2 n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: 10.1155/2008/385362 (2008) for the case mentioned above, and derive the convergence rate n −1/2 β n log 1/2 n for the above β n under the given covariance function, which improves the relevant one n −1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.

Suggested Citation

  • Guodong Xing & Shanchao Yang, 2010. "Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers," Journal of Theoretical Probability, Springer, vol. 23(1), pages 169-192, March.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:1:d:10.1007_s10959-008-0205-3
    DOI: 10.1007/s10959-008-0205-3
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    References listed on IDEAS

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    1. Roussas, G. G., 1994. "Asymptotic Normality of Random Fields of Positively or Negatively Associated Processes," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 152-173, July.
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    3. Roussas, George G., 1995. "Asymptotic normality of a smooth estimate of a random field distribution function under association," Statistics & Probability Letters, Elsevier, vol. 24(1), pages 77-90, July.
    4. Roussas, George G., 1991. "Kernel estimates under association: strong uniform consistency," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 393-403, November.
    5. Oliveira, Paulo Eduardo, 2005. "An exponential inequality for associated variables," Statistics & Probability Letters, Elsevier, vol. 73(2), pages 189-197, June.
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