Randomly stopped sums with consistently varying distributions

Abstract: Let ${\xi_1,\xi_2,\ldots}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. We consider conditions for ${\xi_1,\xi_2,\ldots}$ and $\eta$ under which the distribution function of the random sum $S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of consistently varying distributions. In our consideration, the random variables ${\xi_1,\xi_2,\ldots}$ are not necessarily identically distributed.