Precise Large Deviations For Compound Renewal Risk Model With Dependence Claims

Since the last 1960s, heavy-tailed distributions have been widely used in branching pro-cesses, queueing theory, risk theory and other fields. In the financial and insurance industries, random variables with heavy-tailed distributions have long been regarded as the standard model of the individual claims by a growing number of scholars. In the early researches on insurance and finance, the objects were supposed to be independent, identically distributed random vari-ables. However, in the practical applications, there may exist some dependence among those random variables. And they may be not independent and not identically distributed. So, in this paper, we still regard the random variables with heavy-tailed distributions as the main object, but the difference with the past literature is that the precise large deviations for sums of random vari-ables is focused on the dependence structure and non-identical distributions. In Chapter 2, we discuss the precise large deviations for non-random and random sums of negative dependence random variables with non-identical distribution functions. Under certain conditions, the low-er bound of the precise large deviations for the non-random sums is solved and the uniformly asymptotic results for the corresponding random sums are obtained, and the establishment of a more general different model from the past, and is close to the actual compound renewal risk model. Finally, the research results are applied to the more practical compound renewal risk model, and we obtained precise large deviations for compound renewal risk model and the theo-retical and practical values are verified. In Chapter 3, we deeply discuss the compound renewal risk model, in which we found that the compound renewal risk model can be equivalent to re-newal risk model under certain conditions. While in weaker conditions we obtained asymptotic result in compound renewal risk model. In addition to, these researches also show that the im-pact of this dependency relationship between random variables to precise large deviations of the final result is not significant. In short, all the results we establish extend and improve the related existing results substantially.