Any random variable with right-unbounded distributional support is the minimum of independent and very heavy-tailed random variables

TL;DR

This paper proves that any right-unbounded light-tailed random variable can be represented as the minimum of two independent heavy-tailed variables, extending to multiple variables and discussing potential dependence cases.

Contribution

It establishes a universal representation of light-tailed variables as minima of heavy-tailed variables, generalizing previous specific examples.

Findings

Any light-tailed, right-unbounded variable can be expressed as a minimum of two heavy-tailed variables.

The heavy-tailed variables can have arbitrarily heavy tails.

The result extends to the minimum of any finite number of independent variables.

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 (Leipus et al., AIMS Mathematics, 2023), the authors presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. We will show that this phenomenon is universal: {\it any} light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables. Moreover, a more general fact holds: these two independent random variables may have as heavy-tailed distributions as one wishes. Further, we will extend the latter result onto the minimum of any finite number of independent random variables. We will also comment on possible generalizations of our result to the case of dependent random variables.