Some Discussion Of The Product Of Dependent Random Variables

This dissertation consists of three Parts. We investigate the tail probability of the product of nonnegative dependent random variables from Max-domain of attraction and discuss a discrete-time insurance risk model, in which the insurance risks and financial risks follow jointly multivariate Farlie-Gumbel-Morgenstern distributions.In insurance and finance, the product of two random variables is one of basic elements in stochastic modelling, which has been extensively investigated in lots of works under the assumption that the random variables are independent. However, this assumption is far too unrealistic for most applied problems. Therefore, the study for dependent cases is much important. We assume that non-negative random variables X,Y1,Y2,...,Yn follow distribution from max-domain of attraction of extreme value distributions and their dependence is modeled via a multivariate Farlie-Gumbel-Morgenstern distribution. For each of the Frechet, Gumbel and Weibull cases, we obtain an explicit asymptotic formula for the tail probability of Z. In comparison to the asymptotic formula for the independent case, our results contain the stochastic discount. In addition, we also investigate a discrete-time insurance risk model, in which the insurance risks and financial risks follow jointly multivariate Farlie-Gumbel-Morgenstern distributions.Chapter One is a Preface. It gives an introduction to the development and significance of risk theory’ and the background and main ideas of this dissertation. We also present some knowledge about heavy tails and multivariate Farlie-Gumbel-Morgenstern distribution. In addition, the main construction of this dissertation is presented at the end.Chapter Two investigates the tail probability of the product of finitely many nonnegative dependent random variables. They follow distributions from max- domains of attraction of extreme value distributions and their dependence is modeled via a multivariate Farlie-Gumbel-Morgenstern distribution. For each of the Frechet, Gumbel and Weibull cases, following Breiman (1965) and Hashorva et al.(2010), we obtain an explicit asymptotic formula for the tail probability of the product.In Chapter Three, We consider a discrete-time insurance risk model, in which the insurance risks and financial risks follow jointly multivariate Farlie-Gumbel-Morgenstern distributions. Based on the results in Tang and Tsitsiashvili(2004) and Tang and Vernic (2007), we obtain the asymptotic formulas for the ruin probabilities and give some numerical studies.