Precise Large Deviations For ?-mixing And UND Random Variables With Long-tailed Distributions

The study of precise large deviations theory has a profound significance in insurance. Up until now, many researchers made great efforts in the area of the precise large deviations and had a lot of inspiring results. Heavy-tailed random variables are widely used in insurance and finance models. So combing heavy-tailed distributions and precise large deviations become a hotspot in the research of probability theory, recently. However, in the existing results, little is known about the results for long-tailed distributions. Up to now, there are also some researchers to discuss the properties and applications of ?-mixing sequence,?-mixing sequence are applied into the practical field, such as stationary processes, time series, statistical inference for non-stationary processes and so on. However, to the best of our knowledge, there is little results about ?-mixing sequence in applications of precise large deviations. Most of the researchers al-ways concentrated that the company provides only one kind of insurance contract. In reality this assumption is not correct, so the precise large deviations problem of multi-risk model is more valuable. Moreover, the asymptotic results about long-tailed distributions,?-mixing sequence and multi-risk model are very little, respectively.In this paper, first, we get the stochastic law of large numbers of ?-mixing sequence. Sec-ond, we investigate the precise large deviations for sums of ?-mixing and UND random variables with long-tailed distributions. The asymptotic relations for non-random sum and random sum of random variables with long-tailed distributions are obtained, respectively. The last, in view of the actual condition of the insurance company, a multi-risk model is proposed. As the extension of one-risk model, we investigate the precise large deviations for sums of ?-mixing and UND random variables with long-tailed distributions in a multi-risk model, and obtain the asymptotic relations for non-random sums and random sums of random variables with long-tailed distribu-tions in this model, respectively.