Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum

Abstract

A random variable $\hat{I}¾$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (λξ) <\infty$ for some $\hat{I}»>0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (λξ) = \infty$, for all $\hat{I}»>0$. 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 $\hat{I}{¾}_1$, that allow to find another independent heavy-tailed random variable, say $\hat{I}{¾}_2$, such that their minimum $\min (\hat{I}{¾}_1,\hat{I}{¾}_2)$ is light-tailed. We also provide a number of extensions of this result

17 pages