Extensions Of Breiman’s Theorem And Applications In Risk Theory

As its is known to all, insurance is effective means of transferring and dispersing risk. Risk theory is the theory of mathematical analysis of the potential and unknown risks to the insurance industry. Risk theory is one of the important branches of applied probability theory. It is not only its own important theoretical research value, but also according to the practice of financial insurance to establish a series of risk model, and with probability theory, mathematical statistics and random process as a tool for its mathematical analysis. People have made many important conclusions, and to solve financial and insurance practical problems. The risk theory has been put forward for a hundred years, but it has been introduced into our country for only a few decades. In the recent research of risk theory, the majority of scholars and financial insurance companies explore how to measure the size of the insurance company facing the size of the risk of bankruptcy. They have become one of the core issues of their common study that ruin probability of the expression form and asymptotic behavior are obtained.At present, the research on the theory of ruin probability has a lot of literature. In this paper, we study the properties of the product of random variables based on the theory of ruin probability. In this paper, we first study the tail properties of the product of X and Y, which are independent of each other. We relax the existence of the(a(10)e) th moment of Y in Breiman’s Theorem to EYa. Of course, we impose constrains to the slowly varying functionl, for some β>1:Thereby, the similar Breiman theorem is obtained when the random variables X and Y are independent, and then it is applied to the stochastic equations:R~D=MR+Q;When the random variables X and Y are dependent, and YX),( obey the copula distribution function, the similar Breiman theorem is obtained. When the random vectors YX),( are dependent, the similar Breiman theorem is applied to the discrete time ruin probability model. It depicts the manifestation form of the finite time and infinite time ruin probability.