Extensions Of Breiman’s Theorem Of Product Of Dependent Random Variables With Applications To Ruin Theory

As a core content of risk theory, bankruptcy theory plays a more and more important character in insurance and financial field. In the research of bankrupt-cy theory, firstly we should consider the propertise of the product of random variables. The product of independent random variables has been extensively in-vestigated, but obviously,this assumption may be restrictive in practical context, so the research for dependent cases bacomes more and more important. In this paper, we consider the tail behavior of the product of two dependent random variables X and (?). We relax the existence of the (α+∈)th moment of (?) in Breiman’s theorem to E(?)α, and obtain the similar result as Breiman’s theorem of the dependent product X(?) while X and O follow a copula function. We also show the second-order form of Breiman’s theorem. As applications, we consid-er two kinds of discrete-time insurance risk models, and derive the asymptotic tail behaviors for the ruin probabilities when insurance and financial risks follow jointly the copula function or multivariate Sarmanov distributions.