On Upper Bounds for the Tail Distribution of Geometric Sums of Subexponential Random Variables

TL;DR

This paper refines the method for bounding the tail distribution of geometric sums with subexponential variables, resulting in tighter bounds and practical implementation guidance.

Contribution

It introduces improved probabilistic techniques for upper bounds, correcting previous methods and enhancing their tightness for subexponential sum tail analysis.

Findings

Tighter upper bounds for tail distributions are achieved.

Theoretical improvements are demonstrated through multiple examples.

Implementation strategies for the bounds are provided.

Abstract

The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.