School of Mathematical Science, Soochow University, Department of Applied Mathematics, University of Science and Technology of Suzhou

This paper obtains some asymptotics for the tail probabilities of maximum of sums, random sums and maximum of idetically distributed, negatively associated random variables with heavy tails. The obtained results weaken the conditions of the moments of Theorem 2.1 in Wang and Tang (Statist. Prob. Lett., 68, 287--295, 2004). We also discuss the conditions of Theorem 2.2 of [1] and remove the restrictions of the supports of random variables.