Asymptotic Behavior Of Local Precise Large Deviations

With further study of the precise large deviations and their applications in the field of insurance, more and more researchers are engaged in this topic. For example, the large deviations for nonrandom sums and random sums in the single risk model or in the multi-risk models. Recently, many results began to local precise large deviations theory, for example, local limits theorem for one-sided large precise deviations and the local probabilities for random sums. However, most studies are based on the global large deviations. The research about local precise large deviations is not very perfect, especially about local precise large deviations for random sums. So, we will give them in this paper, i.e. asymptotic behavior of local precise large deviations for random sums, which is based on the non-random sums results of them.We introduce the background of large deviations and some definitions in chapter one. In chapter two. it gives some necessary propositions, lemmas and its proofs. In chapter three, we prove the theorem for asymptotic behavior, when the density function of random variable is consistently varying density. The proof will be divided into three parts and be proved in three lemmas.