A new class of large claim size distributions: Definition, properties, and ruin theory

TL;DR

This paper introduces a new class of large claim size distributions, al J, which generalizes existing classes and retains key properties, with applications to ruin theory and infinitely-divisible distributions.

Contribution

The paper defines the class al J, explores its properties, relations to other classes, and applies it to ruin theory, extending classical results like the Pakes-Veraverbeke-Embrechts theorem.

Findings

al J is more general than subexponential distributions.

Properties of al J are stable under tail-equivalence and convolution.

Partial analogue of ruin theorem established for al J.

Abstract

We investigate a new natural class $\mathcal{J}$ of probability distributions modeling large claim sizes, motivated by the `principle of one big jump'. Though significantly more general than the (sub-)class of subexponential distributions $\mathcal{S}$, many important and desirable structural properties can still be derived. We establish relations to many other important large claim distribution classes (such as $\mathcal{D}$, $\mathcal{S}$, $\mathcal{L}$, $\mathcal {K}$, $\mathcal{OS}$ and $\mathcal{OL}$), discuss the stability of $\mathcal{J}$ under tail-equivalence, convolution, convolution roots, random sums and mixture, and then apply these results to derive a partial analogue of the famous Pakes-Veraverbeke-Embrechts theorem from ruin theory for $\mathcal{J}$. Finally, we discuss the (weak) tail-equivalence of infinitely-divisible distributions in $\mathcal{J}$ with their L\'{e}vy measure.