A Kind Of Precise Large Deviations For Random Variables With Heavy Tails

It is well known that large deviation theory is one of the hot research issues in applied probability theory. Large deviation theory is divided into two parts-precise large deviation and rough large deviation. To the former, scholars have paid great attentions to expand the distribution classes of random variables, ensuring that good characters of large deviation probability has been kept. Recently, some articles have got large deviations for random variables with consistently varying tails in equivalent forms. Such as Ng et al. (2004) obtained precise large deviations for non-negative in-dependent and identically distributed(i.i.d) random variables. Tang (2006) extended that results for negatively orthant dependent random variables. Then, Chen and Zhang (2007) extended his results in the case of random sums. On the other hand, Wang et al. (2006) got precise large deviations with dominatedly varying tails in the case of weak equivalence. Thus, it is worthy to consider that how to establish compatible precise large deviations in the equivalent forms with the extented distribution classes of consistently varying tails. Wang and Wang(2009) obtained precise large devia-tions for negative associated variables with both long tails and dominatedly varying tails((?)∩(?)). In chapter 2 of this paper, we improved their results on extended neg-atively orthant dependent variables while weaken a required condition of that paperFurthermore, we focused on precise large deviations for i.i.d random variables with strong subexponential tails. In chapter 3, we discussed claims in insurance contracts with distributions in (?)∩(?), and obtained precise large deviations for sums of claims in a multi-risk model.