## New examples of heavy-tailed O-subexponential distributions and related closure properties.(English)Zbl 1242.60014

Summary: Let $${\mathcal L}$$ and $${\mathcal S}$$ denote the classes of distributions with long tails and subexponential tails respectively. Let $${\mathcal O}{\mathcal S}$$ denote the class of distributions with $$O$$-subexponential tails, which means the distributions with the tails having the same order as the tails of their 2-fold convolutions. In this paper, we first construct a family of distributions without finite means in $${\mathcal L}\cap{\mathcal O}{\mathcal S}\setminus{\mathcal S}$$. Next some distributions in $${\mathcal L}\cap{\mathcal O}{\mathcal S}\setminus{\mathcal S}$$, which possess finite means or even finite higher moments, are also constructed. In connection with this, we prove that the class $${\mathcal O}{\mathcal S}$$ is closed under minimization of random variables. However, it is not closed under maximization of random variables.

### MSC:

 6e+06 Probability distributions: general theory
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### References:

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