Randomly stopped sums with consistently varying distributions

Abstract

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. We consider conditions for $\{{\xi }_1,{\xi }_2,\dots \}$ 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 $\{{\xi }_1,{\xi }_2,\dots \}$ are not necessarily identically distributed.

Published at http://dx.doi.org/10.15559/16-VMSTA60 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)