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

摘要

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. Keywords Positively associated random variable - Exponential inequality - Rate of convergence Mathematics Subject Classification (2000) 60F15 - 62G20