Lower limits for distributions of randomly stopped sums

TL;DR

This paper investigates the lower bounds of the ratio between the tail distribution of a randomly stopped sum and the original distribution, providing insights into tail behavior when summing a random number of i.i.d. variables.

Contribution

It establishes lower limits for the tail ratio of randomly stopped sums, advancing understanding of tail distribution behavior in stochastic sums.

Findings

Derived explicit lower bounds for tail ratios

Analyzed tail behavior for various classes of distributions

Provided theoretical results applicable to risk and queueing models

Abstract

We study lower limits for the ratio $\frac{\bar{F^{*\tau}}(x)}{\bar F(x)}$ of tail distributions where $ F^{*\tau}$ is a distribution of a sum of a random size $\tau$ of i.i.d. random variables having a common distribution $F$, and a random variable $\tau$ does not depend on summands.