Heavy tails and dependence with applications in insurance

In this thesis we study the tail behavior of a random variable and sum of dependent random variables using the extreme value theory. We examine the tail behavior of a single random variable by mixture distribution models, and the asymptotic properties of the value-at-risk measure of dependent regularly varying random variables. In order to obtain a flexible fit not only on the tail but also on the body of the underlying distribution, mixture distributions are introduced with finite or infinite number of thresholds, where the consistency of the heavy-tailedness is preserved by the conditional layer mixture. Hazard rate functions of the conditional layer mixture distributions are studied and the mixture of the hazard rate functions can be used in modeling the mixture distributions equivalently. Impact of heavy-tailedness and dependence on the value-at-risk measure is examined for the sum of regularly varying random variables under quite general dependence structure and we conclude that the extreme value index completely determines the tail behavior of the compound sum of regularly varying random variables with respect to the value-at-risk measure. In addition, a hierarchical structure composed of maximal Markov sequences is introduced to simplify a given pool of risks under arbitrary dependence and we propose a computational method of the aggregate distribution of each maximal Markov sequence