Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum
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| Format: | Preprint |
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2026
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| author | Foss, Sergey Scheutzow, Michael Tarasenko, Anton |
| author_facet | Foss, Sergey Scheutzow, Michael Tarasenko, Anton |
| contents | A random variable $ξ$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (λξ) <\infty$ for some $λ>0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (λξ) = \infty$, for all $λ>0$. 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 $ξ_1$, that allow to find another independent heavy-tailed random variable, say $ξ_2$, such that their minimum $\min (ξ_1,ξ_2)$ is light-tailed. We also provide a number of extensions of this result |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06452 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum Foss, Sergey Scheutzow, Michael Tarasenko, Anton Probability 60E05, 60F99 A random variable $ξ$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (λξ) <\infty$ for some $λ>0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (λξ) = \infty$, for all $λ>0$. 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 $ξ_1$, that allow to find another independent heavy-tailed random variable, say $ξ_2$, such that their minimum $\min (ξ_1,ξ_2)$ is light-tailed. We also provide a number of extensions of this result |
| title | Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum |
| topic | Probability 60E05, 60F99 |
| url | https://arxiv.org/abs/2603.06452 |