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On the structure of a class of distributions obeying the principle of a single big jump. (English) Zbl 1342.60016

Summary: In this paper, we present several heavy-tailed distributions belonging to the new class \(\mathcal{J}\) of distributions obeying the principle of a single big jump introduced by S. Beck et al. [Bernoulli 21, No. 4, 2457–2483 (2015; Zbl 1362.60007)]. We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class \(\mathcal{J}\) are automatically strongly heavy-tailed and thus have tails which are not too irregular. Second, we show that such distributions are not necessarily weakly tail equivalent to a subexponential distribution. We also show that the class of heavy-tailed distributions in \(\mathcal{J}\) which are neither long-tailed nor dominatedly-varying-tailed is not only non-empty but even quite rich in the sense that it has a non-empty intersection with several other well-established classes. In addition, the integrated tail distribution of some particular of these distributions shows that the Pakes-Veraverbeke-Embrechts theorem for the class \(\mathcal{J}\) does not hold trivially.

MSC:

60E05 Probability distributions: general theory
60G50 Sums of independent random variables; random walks

Citations:

Zbl 1362.60007
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