Large deviations for random sums of negatively dependent random variables with consistently varying tails

Let {Xk,k=1,2,…}{Xk,k=1,2,…} be a sequence of negatively dependent random variables with common distribution F   and finite expectation μμ. Under the assumption that the tail probability F¯(x)=1-F(x) is consistently varying as x   tends to infinity, this paper investigates precise large deviations for the random sum SN(t)=∑n=1N(t)Xn, where {N(t),t⩾0}{N(t),t⩾0} is a nonnegative and integer-valued process independent of {Xk,k=1,2,…}{Xk,k=1,2,…}.