Conditional Distribution of Heavy Tailed Random Variables on Large Deviations of their Sum

TL;DR

This paper investigates the distribution of heavy-tailed subexponential random variables during large deviations of their sum, revealing how individual variables contribute to these rare events.

Contribution

It provides a detailed analysis of the distributional behavior of subexponential variables under large sum deviations, extending understanding beyond the single-variable deviation principle.

Findings

Large deviations are primarily caused by a single variable's deviation.

Distributional structure during large deviations is characterized in detail.

Insights into the behavior of heavy-tailed sums in rare event regimes.

Abstract

It is known that large deviations of sums of subexponential random variables are most likely realised by deviations of a single random variable. In this article we give a detailed picture of how subexponential random variables are distributed when a large deviation of their sum is observed.