ON THE STRUCTURE OF A CLASS OF DISTRIBUTIONS OBEYING THE
PRINCIPLE OF A SINGLE BIG JUMP
Hui Xu
Michael Scheutzow
Yuebao Wang
Zhaolei Cui
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.