Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum

TL;DR

This paper investigates conditions under which the minimum of two heavy-tailed random variables can be light-tailed, providing a theoretical framework for understanding tail behavior transformations.

Contribution

It establishes necessary and sufficient conditions for heavy-tailed variables to produce a light-tailed minimum, extending previous work on tail behavior representations.

Findings

Identifies conditions for heavy-tailed variables to have a light-tailed minimum

Provides a theoretical characterization of tail behavior transformations

Extends previous results on tail distributions and minima

Abstract

A random variable $\xi$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (\lambda \xi) <\infty$ for some $\lambda >0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (\lambda\xi) = \infty$, for all $\lambda>0$. In \cite{LSK1}, the authors presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. In \cite{FKT}, it was shown that any light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables, with further generalisations of the result in a number of directions. We analyse an ``inverse'' question. Namely, we obtain necessary and sufficient conditions on the distribution of a heavy-tailed random variable, say $\xi_1$, that allow to find another independent heavy-tailed random variable, say $\xi_2$, such that their minimum $\min (\xi_1,\xi_2)$ is light-tailed. We also provide a number of extensions of this result