Viewpoints on Large Deviations of Multivariate Subexponential Distributions

Insurance supervision demands that the ruin probability of an insurance company does not exceed a certain level. To compute these ruin probabilities, insurance companies model inter alia the cumulative payout process as a multivariate random walk, where the different components represent different lines of insurance. In particular, in non-life insurance, those random walks can have heavy-tailed increments. Companies are interested in the asymptotic behaviour of multivariate random walks because it provides useful information, such as probability bounds, for ruin probabilities. This licentiate thesis presents different viewpoints on large deviations of such multivariate random walks with subexponentially distributed increments. After introducing the concept of subexponentiality and its main properties, we derive the asymptotic behaviour of a random walk with generalized Weibull distributed increments. This generalized Weibull distribution belongs to the class of stretched exponential distributions, which are heavy-tailed distributions but have finite moments of all orders. Then, we extend subexponentiality to a multivariate setting. Therefore, we define a one-dimensional distribution as the probability that the random vector belongs to a shifted increasing set. Furthermore, we study the asymptotic behaviour of large deviation probabilities of the sum of subexponentially distributed random vectors. We generalize the result for random vectors with independent components to a binomial model and give sufficient conditions for the principle of a single big jump to hold in each component. Finally, we examine the ruin probabilities for multivariate random walks with subexponentially distributed increments. Here, we study a model with discrete time and compute the asymptotic behaviour of the ruin probability