  Beck et al. (2013) introduced a new distribution class J which contains many
heavy-tailed and light-tailed distributions obeying the principle of a single
big jump. Using a simple transformation which maps heavy-tailed distributions
to light-tailed ones, we find some light-tailed distributions, which belong to
the class J but do not belong to the convolution equivalent distribution class
and which are not even weakly tail equivalent to any convolution equivalent
distribution. This fact helps to understand the structure of the light-tailed
distributions in the class J and leads to a negative answer to an open question
raised by the above paper.
