Precise Large Deviations Of Two Non-standard Renewal Risk Models With Heavy-tailed Claims

Heavy-tailed analysis is one of the branches of extreme value theory which can be extensively applied to risk management in insurance. In recent years, extreme events have occurred frequently, such as hurricane, earthquake, finan-cial crisis and so on. Once these extreme events occurred, they may often lead to a huge loss and even the bankruptcy of the insurer directly. The study on applied probability has shown that heavy-tailed distributions can be used to model the catastrophic risk resulted by these extreme events in the risk theory. They also play a key role in some fields such as insurance, financial mathematics and queueing theory.The study on precise large deviations for random sums with heavy-tailed claims is an important topic in the fields of heavy-tailed distributions. Its results can be used to estimate ruin probability. Some classic work on precise deviations with heavy tails can be found in Heyde(1967a), Heyde(1968) and Nagaev(1969). However, these existing results were based on the inde-pendence of claim variables. Note that there are many types of dependence structures existing in insurance reality. The study on precise large deviations has received a remarkable amount of attention recently. The reader is referred to Tang et. al.(2001) [33], Liu(2009), Wang and Wang(2013) for some recent works. Liu(2009) obtained the precise large deviations for partial sums for END random variables with consistently varying tails, and Wang and Wang(2013) investigated the precise large deviations for random sums for END random variables with consistently varying tails.Based on some previous works, consider two nonstandard renewal risk models in this paper. Under the assumption that claim variables satisfy some certain dependence structures, the precise large deviations for random sums with heavy-tailed claims are obtained. The main contributions of this paper are the following aspects.Firstly, let{Xk,k≥1} be a sequence of END real-valued random variables with common distribution, and{N(t),t≥0} be a nonnegative integer-valued counting process independence of{Xk, k≥1}. Consider random sums SNt,c= where c is a real number. Under Assumptions A and B, Theorem 2.1 is obtained, which is the precise large deviations of random sums in the presence of END structure with dominated variation. Theorem 2.1 extends the works of Liu(2009) and Wang and Wang(2013).Secondly,let{Xk,k≥1} and{θk, k≥1} be claim sizes and inter-arrival times, respectively. Suppose that{(Xk,θk), k≥1} form a sequence of indepen-dent and identically distributed (i.i.d.) random vectors (there is a dependence structure between Xk and θk). Assume that {Zn,n≥1} constitute anoth-er sequence of identically distributed positive integer-valued random variables independent of{(Xk,θk), k≥1}, which denote the real claim times in the nth event. the renewal counting process and the compound renewal counting process, re-spectively. Then forms a compound renewal risk model and the model becomes a regression-type size-dependence compound renewal risk model under the Assumption 3. As for this model, Theorem 3.1 is obtained, which is the precise large deviations of a compound renewal risk model with regression-type size-dependence structure. Theorem 3.1 extends the work of Bi and Zhang(2013).