| Abstract [eng] |
In the first part of the thesis, we consider the following problem. Let \{ξ_1,ξ_2,…\} be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for random variables \{ξ_1,ξ_2,…\} and η under which distribution function of random sum S_η=ξ_1+ξ_2+⋯+ξ_η belongs to the class of consistently varying distributions. In our consideration random variables \{ξ_1,ξ_2,…\} are not necessarily identically distributed. In the second part of the thesis, we consider the following problem. Let \{ξ_1,ξ_2,…\} be a sequence of independent real-valued, possibly nonidentically distributed, random variables, and let η be a nonnegative, nondegenerate at 0, and integer-valued random variable, which is independent of \{ξ_1,ξ_2,…\}. We consider conditions for \{ξ_1,ξ_2,…\} and η under which the distributions of the randomly stopped minimum, maximum, and sum are regularly varying. In the third part of the thesis, we consider the sum S_n^ξ=ξ_1+⋯+ξ_n of possibly dependent and non-identically distributed real-valued random variables ξ_1,…,ξ_n with consistently varying distributions. By assuming that collection \{ξ_1,…,ξ_n \} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for the left truncated moments of random sums. The main results of the thesis were proved using the classical methods of probability theory and mathematical analysis. |
