Xu, Hui; Scheutzow, Michael; Wang, Yuebao; Cui, Zhaolei On the structure of a class of distributions obeying the principle of a single big jump. (English) Zbl 1342.60016 Probab. Math. Stat. 36, No. 1, 121-135 (2016). 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. Cited in 8 Documents MSC: 60E05 Probability distributions: general theory 60G50 Sums of independent random variables; random walks Keywords:principle of a single big jump; strongly heavy-tailed distribution; weak tail equivalence; integrated tail distribution Citations:Zbl 1362.60007 PDFBibTeX XMLCite \textit{H. Xu} et al., Probab. Math. Stat. 36, No. 1, 121--135 (2016; Zbl 1342.60016) Full Text: arXiv Link