5.2.1. Pareto

This distribution is c.m., as its ccdf *FX*(*x*)=(<sup>1</sup> + *bu*)−*<sup>a</sup>* can be written as the LST of the Gamma distribution with shape and scale parameters *a* and *b*, respectively, i.e.,

$$(1+bu)^{-a} = \int\_0^{+\infty} e^{-uy} \frac{y^{a-1}}{\Gamma(a)b^a} e^{-y/b} dy.$$

The *n*th moment of the Pareto distribution exists if and only if the shape parameter is greater than *n*. As we are interested in comparing the spectral approximation to the asymptotic approximation of Section 4, it is necessary to have a finite first moment for the claim sizes. Therefore, the shape parameter *a* must be chosen to be greater than 1.

Using Proposition 1, we can easily verify that

$$dS\_H(y) = \frac{1}{\phi c^2} \left( \frac{\nu\_1 \nu\_2 - \varepsilon \rho\_1 \left(\theta \nu\_1 + (1 - \theta)\nu\_2\right)}{-(y + \rho\_1)\rho\_1} + \frac{\nu\_1 \nu\_2}{y\rho\_1} \right) \frac{y^{a-1}}{\Gamma(a)b^a} e^{-y/b} dy.$$
