Let $\{��_1,��_2,\ldots\}$ be a sequence of independent random variables, and $��$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=��_1+��_2+\cdots+��_n$ for $n\geqslant1$. We consider conditions for random variables $\{��_1,��_2,\ldots\}$ and $��$ under which the distribution functions of the random maximum $��_{(��)}:=\max\{0,��_1,��_2,\ldots,��_��\}$ and of the random maximum of sums $S_{(��)}:=\max\{S_0,S_1,S_2,\ldots,S_��\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{��_1,��_2,\ldots\}$ are not necessarily identically distributed.

Published at http://dx.doi.org/10.15559/17-VMSTA74 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)