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

Abstract

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation µ > 0. 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 both the partial sums S n and the random sums S N (t) , where N (t) is a counting process independent of the sequence {X k , k ≥ 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.

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