Precise large deviations for sums of random variables with consistently varying tails

Let {X-k, k greater than or equal to 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation mu > 0. Under the assumption that the tail probability (F) over bar (x) = 1 - F(x) is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S, and the random sums S-N(t), where N((.)) is a counting process independent of the sequence {X-k, k greater than or equal to 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.