Precise Large Deviations And Asymptotics Of The Tail Probability For Sums Of Negatively Associated Random Variables With Heavy Tails

Since the 1960s, heavy-tailed distributions have been widely used in branching processes, queueing theory, risk theory including insurance and finance and other fields. In the early researches on insurance and finance, the objects were supposed to be independent, identically distributed random variables. However, in the practical applications, there may exist some dependence among these random variables. And they may not be independent. So, in this paper, we still regard the heavy-tailed distributions as the main object and discuss the precise large deviations and asymptotics of the tail probability for sums of negatively associated random variables. In Chapter 2, we discuss the precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, the three precise large-deviation probabilities with different centering numbers are equivalent to each other. On the basis of the study, we consider precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve corresponding results of Ng et al.(2004)~[31] In Chapter 3, we obtains some asymptotics for the tail probabilities of maximum of sums of idetically distributed, negatively associated random variables. The obtained results improve some corresponding results of Wang and Tang(2004)~[35].