Precise Large Deviations For Sums Of Widely Orthant Dependent Random Variables

Since the pioneering works of C.C.Heyde, A.V.Nagaev and S.V.Nagaev in 1960's and 1970's, the precise asymptotic behavior of large-deviation probabilities of sums of heavy-tailed random variables has been extensively investigated by many people, but mostly it is assumed that the random variables under discussion are independent. In this article, we set up dependent multi-risk models in insurance, where all the claims constitute WOD (widely orthant depen-dent) random arrays. Precise large deviations for non random sums and random sums of the dependent multi-risk models with WOD structure and consistent variation are investigated. The obtained results extend those of Wang and Wang (2007), Liu (2009), Chen et al.(2011), Wang and Wang(2013).The contents of the paper is as follows. Section 1 is the introduction, will introduce basic knowledge and prepare lemma involved in this paper. Meanwhile, provide theoretical basis for the proof of theorem. Section 2 We have given the theorem and the process of theorem proving of precise large deviations for two-dimensional and multidimensional non random sums. Section 3 We have given the theorem and the process of theorem proving of precise large deviations for two-dimensional and multidimensional random sums. Section 4, we provide support for the theorem through the numerical simulation. Section 5 ends the paper with a brief discussion and expectation.