Heavy-tailed Dependent Random Variables And The Asymptotic Distribution

As the central branch of ruin theory, ruin probability of heavy-tailed distributions is a hot topic in risk theory. Since the pioneering works of C. C. Heyde, and S. V. Nagaev in 1960's and 1970's (see [1][2]), the precise asymptotic behavior of large deviation prob-abilities of sums of heavy-tailed random variables have been extensively investigated by mang people, but mostly it is assumed that the random variables under discussion are independent. In this paper, several cases of dependent random variables and the asymp-totics tail probabilities of their partial sums are discussed.The article is divided into four chapters according to contents:Chapter 1 is preface. We review the development and the current situation of sums of the heavy-tailed distributions, then introduce some related theories of heavy-tailed dis-tributions and copula functions.Chapter 2 stuties the linear Spearman copula dependent heavy-tailed random vari-ables and the asymptotics tail probabilities of their partial sums. First, the relations of two random variables are obtained:Let X1, X2 be the positive linear Spearman copula dependent random variables with distributions Fj∈(?) satisfying Fi(-x)= o(Fi(x)), for i= 1,2. Then:Let X1, X2 be the negative linear Spearman copula dependent random variables with distributions Fi∈(?) satisfying Fi(-x)= o(Fi(x)), for i= 1,2. Then:On the basis, the relations of n positive linear Spearman copula dependent random vari-ables are obtained: The relations of n negative linear Spearman copula dependent random variables are also obtained: whereIn Chapter 3, the Ali-Mikhail-Haq copula dependent heavy-tailed random variables are discussed. We get the relations:Let X1, X2,…, Xn be the Ali-Mikhail-Haq copula dependent random variables with distributions Fi∈(?) satisfying Fi(-x)= o(Fi(x)), for i= 1,2,…, n. Then:Let X1, X2,…, Xn on [0,∞) be the Ali-Mikhail-Haq copula dependent random vari-ables with distributions Fi∈(?), i= 1,2,…, n. Then:In Chapter 4, the application of Chapter 2 and Chapter 3 in the risk theory is considered. We prove that it is feasible to calculate the ruin probabilities.