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

Abstract: A random variable X has a light-tailed distribution (for short: is light-tailed) if it possesses a finite exponential moment, E \exp (cX) is finite for some c>0, and has a heavy-tailed distribution (is heavy-tailed) if E \exp (cX) is infinite, for all c>0. Leipus, Siaulys and Konstantinides (2023) 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 one wishes. Further, we extend the latter result onto the minimum of any finite number of independent random variables.