## Bernoulli

• Bernoulli
• Volume 21, Number 4 (2015), 2457-2483.

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

#### 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 Lévy measure.

#### Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 2457-2483.

Dates
Revised: February 2014
First available in Project Euclid: 5 August 2015

https://projecteuclid.org/euclid.bj/1438777601

Digital Object Identifier
doi:10.3150/14-BEJ651

Mathematical Reviews number (MathSciNet)
MR3378474

Zentralblatt MATH identifier
1362.60007

#### Citation

Beck, Sergej; Blath, Jochen; Scheutzow, Michael. A new class of large claim size distributions: Definition, properties, and ruin theory. Bernoulli 21 (2015), no. 4, 2457--2483. doi:10.3150/14-BEJ651. https://projecteuclid.org/euclid.bj/1438777601

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