Strong Limit Theorems For A Class Of Dependent Heavy-tailed Random Sequence And Its Applications

Probability theory is a science which studies the regularity of random phenomenon. It has a wide range of applications in the filed of natural sciences and social sciences. Since the1930s, there are emerging a lot of new branches and limit theory of probability theory is one of its main branches. It is a base of applications on probability theory and mathematical statistics. Classical limit theory of random variables main study the limit property when its expectations don’t exist. But there are not all of random variables with an finite mean. So heavy-tailed distribution is one of the core research questions of risk theory. The heavy-tailed distribution is more in line with the actual risk relative to the light-tailed.Therefore, this paper studies the strong limit theorem for heavy-tailed (expectation does not exist) dependent random Sequence and its applications. Then we give the ex-tended Borel-Cantelli lemma necessary and sufficient condition of pairwise NQD random sequence on the basis of the limit theorem of pairwise independent random Sequence. Meanwhile, We get Borel-Cantelli lemma and Marcinkiewicz-Zygmund strong law of large numbers of pairwise NQD heavy tailed random sequence. Next, we further promoted on the above results. We consider more general random Sequence that are AQSI random Sequence. Under appropriate conditions, we can get the Borel-Cantelli lemma necessary and sufficient conditions of AQSI random Sequence, and obtain Marcinkiewicz-Zygmund type strong law of large numbers of AQSI heavy-tailed random sequence.At last, we introduce Copula function, and give the equivalent definition of pairwise NQD and AQSI random sequence. Depending on the risk historical data, we simulate different risk by the Copula families of pairwise NQD random Sequence. Then we can predict the future earnings of the different portfolio.