Randomly stopped maximum and maximum of sums with consistently varying distributions

TL;DR

This paper investigates the conditions under which the distributions of the maximum of a sequence of independent, not necessarily identically distributed variables and the maximum of their partial sums, stopped at a random time, belong to the class of consistently varying distributions.

Contribution

It establishes new criteria for the distributional properties of maxima and partial sums of independent, non-i.i.d. variables stopped at a random time, expanding understanding of their tail behaviors.

Findings

Distribution functions belong to consistently varying class under certain conditions.

Results apply to non-identically distributed variables.

Provides criteria for tail behavior of maxima and sums.

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.