# The Stability of the Aggregate Loss Distribution

## Abstract

**:**

## 1. Introduction

## 2. Saddlepoint Approximation to the Stability

#### 2.1. The Sum

**Theorem**

**1.**

**Proof.**

#### 2.2. The Compound Sum

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 3. Numerical Illustrations

`fminsearch`was used for computing the saddlepoint. Matlab’s programs used for these computations are available at http://www.stat.unibe.ch.

#### 3.1. Poisson-Gamma Total Claim Amount

#### 3.2. Geometric-Weibull Total Claim Amount

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Derivatives of the Cumulant Generating Function of the Perturbed Summand

#### Appendix A.2. Derivatives of the Cumulant Generating Function of the Perturbed Compound Sum

## References

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**Figure 1.**Poisson with $\lambda =2$ compound sum of independent gamma with $\alpha =2$ and $\beta =3$ random variables. Dashed curve: s.f. Continuous curve, from lowest to highest curve: approximate stabilities for perturbation points $x=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}10$, respectively.

**Figure 2.**Geometric with $\rho =3/10$ compound sum of independent Weibull with $\alpha =3$ random variables. Dashed curve: s.f. Continuous curve, from lowest to highest curve: approximate stabilities for perturbation points $x=1/2,\phantom{\rule{0.166667em}{0ex}}3/2,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}7$, respectively.

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**MDPI and ACS Style**

Gatto, R.
The Stability of the Aggregate Loss Distribution. *Risks* **2018**, *6*, 91.
https://doi.org/10.3390/risks6030091

**AMA Style**

Gatto R.
The Stability of the Aggregate Loss Distribution. *Risks*. 2018; 6(3):91.
https://doi.org/10.3390/risks6030091

**Chicago/Turabian Style**

Gatto, Riccardo.
2018. "The Stability of the Aggregate Loss Distribution" *Risks* 6, no. 3: 91.
https://doi.org/10.3390/risks6030091