Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables

In this paper, asymptotic lower bounds of precise large deviations for non-random sums and random sums of nonnegative random variables (r.v.s) are derived under some fairly weak conditions. The obtained results are used to derive asymptotic lower bounds of precise large deviations in a multi-risk model. All the results we establish extend and improve the related existing results substantially.