| Abstract [eng] |
In this summary the main results of the dissertation are presented. In the thesis the sums of dependent nonidentically distributed heavy-tailed random variables are investigated. With some conditions for the (heavy-tailed) distribution of maximal element, the weak max-sum equivalence is proved. Some copula-based examples of dependence structures are given. The sums with dependent nonidentically distributed r.v.s and positive random weights are discussed. The closure property of weighted sums is proved. That is, given that marginal distributions are from the long-tailed distribution class, the distribution of sum belongs to the same class. Moreover, asymptotic equivalence of the tail probabilities of the sum and the sum of nonnegative random variables with their weights is shown. It is shown that this result holds if dependence of random variables X1,...,Xn is generated by the well-known FGM copula. Finally, the randomly weighted and stopped dependent sums with identically distributed dependent heavy-tailed r.v.s X1,...,Xn are discussed. The asymptotic lower and upper bounds for the tail distributions of maximum of such randomly stopped sums are derived. Furthermore, the conditions for this result are shown for the wide class of heavy tailed distribution functions and dependence structures. |
