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We establish some exponential inequalities for positively associated random variables without the boundedness assumption. These inequalities improve the corresponding results obtained by Oliveira 2005 . By one of the inequalities, we obtain the convergence rate n −1/2 log log n 1/2 log n 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 and improves the convergence rate n −1/3 log n 5/3 derived by Oliveira 2005 for the above case.