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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:

60E05 Probability distributions: general theory
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