Precise Large Deviations For Sums Of Heavy-tailed Random Variables And Related Problems

Heavy-tailed distributions are one of the core issues in actuarial science, because they are more in accordance with claims' reality than light-tailed ones. Large devia-tion is an important study task in applied probability, and it is usually used to quanti-tatively characterize the property of extremal events. By this reason, recently, precise large deviations for sums of heavy-tailed random variables are widely investigated by many applied probability authors. Based on the existed research results, precise large deviations for single risk models and multi-risk models are studied respectively in the present dissertation, and asymptotic estimate for tail probabilities of randomly weighted pairwise QAI (quasi-asymptotically independent) random variables' sums with applications in risk theory are investigated in the end. The main contents include the following aspects.Firstly, we consider a sequence of UND heavy-tailed random variables belonging to C with common distributions, and there exists some constant c> -∞such that F(c)=1. Under some mild conditions, we prove precise large deviations for determi-nate sums and random sums respectively, and the obtained results extend some classic ones(Tang(2006), Liu(2007), Chen et al.(2007)).Secondly, since all the existed precise large deviations holds only in the heavy-tailed subclass of C so far, we discuss a sequence of NA heavy-tailed random variables in D∩L with common distributions, by using the property of "h-insensitive" function, under some conditions, we get precise large deviations for determinate sums and ran-dom sums, and this is the first time to extend precise large deviations to some larger heavy-tailed subclasses.Thirdly, noting that single insurer usually manages many different types of con-tracts at the same time in reality, we consider multi-risk models. Let {Xij, i=1,…,k,j≥1} be an independent random matrix, and for anyi=1,…,k, Fi∈C. Under some assumptions, we prove precise large deviations for two indices sums with determinate ones∑i=1k∑j=1ni Xij and random ones∑i=1k∑j=1Ni(t) Xij, where{Ni(t),i=1,…,k} be a se-quence of independent renewal counting process, independent of {Xij, i=1,…,k,j≥ 1}. This is also the first time to obtain precise large deviations for multi-risk model with heavy tails, and the obtained results is an extension for ones in single risk model.Fourthly, based on the previous results, let{Xij,i= 1,…,k, j> 1} be a NA ran-dom matrix, and for any i=1,…, k, Fi∈C. Under some conditions, we also get pre-cise large deviations for determinate sums∑i=1k∑j=1ni Xij and random sums∑i=1k∑j=1Ni(t) Xij. The obtained results indicate precise large deviations are also insensitive to NA depen-dent structure in multi-risk models.Finally, we investigate uniformly asymptotic estimate for tail probabilities of ran-domly weighted pairwise QAI random variables' sums and asymptotic formula of ruin probabilities for non-classic continuous time renewal risk model with constant interest force and constant premium rate, where we suppose claims are a sequence of pairwise QAI random variables.