Abstract: In this paper, we present several heavy-tailed distributions belonging to the new class of distributions obeying the principle of a single big jump introduced by Beck et al. (2015). We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class 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 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 does not hold trivially.

2000 AMS Mathematics Subject Classification: Primary: 60E05; Secondary: 60G50.

Keywords and phrases: Principle of a single big jump, strongly heavy-tailed distribution, weak tail equivalence, integrated tail distribution.