On a transformation between distributions obeying the principle of a single big jump

Beck et al. (2013) introduced a new distribution class JJ 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 JJ 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 JJ and leads to a negative answer to an open question raised by the above paper.