Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum

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