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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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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.

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Correspondence to Guodong Xing.

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Xing, G., Yang, S. Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers. J Theor Probab 23, 169–192 (2010). https://doi.org/10.1007/s10959-008-0205-3

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