  In this paper, we present several heavy-tailed distributions belonging to the
new class J of distributions obeying the principle of a single big jump
introduced by Beck et al. [1]. We describe the structure of this class from
different angles. First, we show that heavy-tailed distributions in the class 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 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 nonempty 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 J in [1] does not hold trivially.
