Tail Probabilities Of Dependent Risk Models For Heavy-tailed Random Variables

This dissertation focuses on discussion of the tail behavior of (dependent)risk models under heavy-tails. The main contents include the following aspects.Firstly, we consider a sequence of negatively dependent random variables{X_k,k≥1} with common distribution F and mean 0 and a random weightssequence {θ_k,k≥1} satisfying P(a≤θ_k≤b)=1 for some 0＜a≤b＜∞,k≥1. Under some mild conditions we obtain the precise large deviations for therandomly weighted sum (?).Secondly, following Ng et al. (2003), we propose a customer-arrival-basedinsurance risk model, in which customer's actual claim sizes are described as independentidentically distributed heavy-tailed random variables multiplied witha generalized shot function, and the model can be treated as a generalized Poissonshot noise process. We obtain some precise large deviation results for theactual loss process and extend the results to the multi-risk model.Thirdly, we obtain the uniform estimate for discounted aggregate claims inthe continuous-time renewal model of upper-tailed independent and heavy-tailedrandom variables. With constant interest force and constant premium rate, weestablish a uniform simple asymptotic formula for ruin probability of the renewalmodel in the case that the initial surplus is large.Fourthly, we obtain the uniform estimate for maximum of sums of uppertailindependent and heavy-tailed random variables with nonnegative dependentrandom weights. Then the applications to ruin probabilities in a discrete timerisk model with dependent gross losses and dependent stochastic returns areconsidered.Finally, we consider the random sums of one type of asymptotically quadrantsub-independent and identically distributed random variables {X,X_i,i=1,2,…} with consistent variation tails. We obtain the asymptotic behavior ofthe tail P(X₁+…+X_η＞x) under different cases of the interrelationships between the tails of X andη, whereηis an integer-valued random variable independentof {X,X_i,i=1,2,…}. We state some applications of the asymptoticresults to ruin probabilities in the compound renewal risk model under dependentrisks. We also state some applications to a compound collective risk model underMarkov environment.