Insensitivity to negative dependence of the asymptotic behavior of precise large deviations

Since the pioneering works of C. C. Heyde, A. V. Nagaev, and S. V. Nagaev in 1960's and 1970's, the precise asymptotic behavior of large-deviation probabilities of sums of heavy-tailed random variables has been extensively investigated by many people, but mostly it is assumed that the random variables under discussion are independent. In this paper, we extend the study to the case of negatively dependent random variables and we find out that the asymptotic behavior of precise large deviations is insensitive to the negative dependence.