Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum

A random variable $\xi$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\text{E}\exp (\lambda \xi )<\mathrm{\infty }$ for some $\lambda >0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\text{E}\exp (\lambda \xi )=\mathrm{\infty }$, 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