Randomly stopped maximum and maximum of 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. In addition, let $S_0:=0$ and $S_n:=\xi_1+\xi_2+\cdots+\xi_n$ for $n\geqslant1$. We consider conditions for random variables ${\xi_1,\xi_2,\ldots}$ and $\eta$ under which the distribution functions of the random maximum $\xi_{(\eta)}:=\max{0,\xi_1,\xi_2,\ldots,\xi_{\eta}}$ and of the random maximum of sums $S_{(\eta)}:=\max{S_0,S_1,S_2,\ldots,S_{\eta}}$ belong to the class of consistently varying distributions. In our consideration the random variables ${\xi_1,\xi_2,\ldots}$ are not necessarily identically distributed.