On Sums of Conditionally Independent Subexponential Random Variables

TL;DR

This paper investigates the tail behavior of sums of dependent subexponential random variables, introducing new conditions and concepts like boundary classes to extend understanding beyond independent cases.

Contribution

It develops a framework for analyzing dependent subexponential variables using conditional independence and introduces the boundary class concept.

Findings

Sufficient conditions for tail behavior similar to independent case

Introduction of boundary class as a tool for subexponential analysis

Examples demonstrating dependence effects on tail behavior

Abstract

The asymptotic tail behaviour of sums of independent subexponential random variables is well understood, one of the main characteristics being the principle of the single big jump. We study the case of dependent subexponential random variables, for both deterministic and random sums, using a fresh approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. For a subexponential distribution, we introduce the concept of a boundary class of functions, which we hope will be a useful tool in studying many aspects of subexponential random variables. The examples we give in the paper demonstrate a variety of effects owing to the dependence, and are also interesting in their own right.