Notes on the exponential inequalities for strictly stationary and positively associated random variables

We show some new exponential inequalities for strictly stationary and positively associated random variables being unbounded. These inequalities improve the corresponding results which Sung [2007. A note on the exponential inequality for associated random variables. Statist. Probab. Lett. 77, 1730-1736] got. As application, we obtain the rate of convergence n-1/2(loglogn)1/ψ(logn)2 with any ψ>2 for the case of geometrically decreasing covariances, which closes to the optimal achievable convergence rate for independent random variables under the Hartman-Wintner law of the iterated logarithm, while Sung [2007. A note on the exponential inequality for associated random variables. Statist. Probab. Lett. 77, 1730-1736] only got n-1/3logn for the case mentioned above.