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

TL;DR

This paper explores a transformation that maps heavy-tailed distributions to light-tailed ones within a specific class, revealing new distributions and answering an open question about their structural properties.

Contribution

It introduces a simple transformation that identifies light-tailed distributions in class J not belonging to convolution equivalent classes, clarifying their structure.

Findings

Identifies light-tailed distributions in class J outside convolution equivalent class

Provides a transformation linking heavy-tailed and light-tailed distributions

Answers an open question about the structure of distributions in class J

Abstract

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.