Precise Large Deviation For Sums Of Negatively Dependent Random Variables With Dominatedly Varying Tails

As an important part of the probability theory, large deviation principle is extremely useful in quantitatively describing extremal events. In the early researches, the classical large deviation principles was contributed by Cramer et al. But the random variables which they concerned with had the light-tailed distributions(i.e., the moment functions of the random variables are finite). Since, the heavy-tailed distribution have been investigated widely and deeply under financial insurance realm,and many problems of them come down to one of precise large deviations(e.g. the problems of reinsurance). Therefore.the precise large deviations of partial sums and random sums of heavy-tailed random variables have become a key concerned subject of applied probability researchers.Along with the continuous deepening of application, the independent distribution with the assumption that it was too rational. So people are of great interests towards precise large deviations for heavy tailed random variables satisfying certainly mutually dependence. Wang et al. (2006) established the precise large deviations for non-random and random sums of negatively associated nonnegative random variables with a common dominatedly varying tail distribution function. Tang(2006) gets the precise large deviations for non-random of negatively dependent random variables with a common dominatedly varying tail distribution function on the whole coordinates. In the second chapter, We will as well as promote the latter changes to common dominatedly varying tails and to the random sums results. On the basis of the study, in chapter 3, we also have same results about large deviations for random variables with different distributions. For the conviences of studying the above contents, this paper will introduce heavy-tailed distributions and negatively dependent in the first chapter.