Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails

Let {Xk,k≥1}{Xk,k≥1} be a sequence of negatively dependent random variables with common distribution FF and mean 0. Suppose that F̄(x)=1−F(x) is positive for all xx and consistently varying as x→∞x→∞. Let {θk,k≥1}{θk,k≥1} be another sequence of random variables independent of {Xk,k≥1}{Xk,k≥1} satisfying P(a≤θk≤b)=1P(a≤θk≤b)=1 for some 0