Randomly stopped maximum and maximum of sums with consistently varying distributions

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η 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 $n\geqslant 1$. We consider conditions for random variables $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution functions of the random maximum $\xi _{(\eta )}:=\max \{0,\xi _{1},\xi _{2},\dots,\xi _{\eta }\}$ and of the random maximum of sums $S_{(\eta )}:=\max \{S_{0},S_{1},S_{2},\dots,S_{\eta }\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

Abstract licence: CC BY

A person counts once a day per paper. We keep no cookie and no address: a reader is a hash that changes every day. Any month under five reads shows no number.