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

Abstract A random variable $\xi$ has a light-tailed distribution (for short, is light-tailed) if it possesses a finite exponential moment, ${\mathbb{E}} \, {\exp}{(\lambda \xi)} <\infty$ for some $\lambda >0$ , and has a heavy-tailed distribution (is heavy-tailed) if ${\mathbb{E}} \, {\exp}{(\lambda\xi)} = \infty$ for all $\lambda>0$ . Leipus et al. (2023 AIMS Math. 8 , 13066–13072) presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. We show that this phenomenon is universal: 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 we wish. Further, we extend the latter result to the minimum of any finite number of independent random variables. We also comment on possible generalizations of our result to the case of dependent random variables.