Independent Product Of Heavy-tailed Traits And Their Applications In Risk Theory

The dissertation discusses behaviors of the product of independent random variables and their applications in finance and insurance risk theories under the condition that the distributions are with heavy tails.The product of independent random variables (independent product) plays an important role in the finance and insurance risk theories. By considering the stochastic economic factors such as interest rate and interest force into the classical risk model, the model becomes even more complicated and often includes independent products. Most problems in the risk theory are investigated under the assumption that the claim size distribution is heavy-tailed, so we need to know on what conditions heavy-tailed class has the tail stability, that is, independent product XY does not change dramatically in distributions, especially in the tails, compared with the original X. This tail stability requirement is important for setting up polices against extremal events. We also need to study on what conditions independent product of light-tailed random variables is heavy-tailed. The behaviors of independent product can not be investigated simply by taking logarithm to make independent product into independent sum, thus we need to study the behaviors of independent product in another way.We study the stabilities of heavy-tailed classes especially for class ￡ and class S. For class L, if Y is not degenerate at 0, then the distribution of any continuous random variable X belonging to ￡ multiplied by Y still in the class L; If X is not continuous, independent product XY belongs to class ￡ when Y satisfies some condition. For class S, our results extend Cline and Samorodnitsky (1994)'s significantly. As for other heavy-tailed classes, we prove that if X belongs to one of classes V, A~*, M and M~*, then XY and X are in the same distribution class under some mild conditions on Y, i.e., classes V, A~*, M and M~* have the desired stabilities.For some problems about the independent product of light-tailed random variables, we obtain some sufficient conditions that the distribution of the independent product of two light-tailed random variables belongs to class ￡ and class M, such as, the distribution of any continuous random variable X belonging to L(γ) multiplied by an unbounded Y belongs to class L; the distribution of any random variable X belonging to L(γ) multiplied by an unbounded Y belongs to class M. We also give another sufficient condition...