*5.1. Interclaims Times*

We choose a hyperexponential distribution with two phases, i.e., *K* ∼ *<sup>H</sup>*2(*<sup>θ</sup>*, 1 − *θ*; *ν*1, *<sup>ν</sup>*2), such that ˜ *k*(*s*) = *ν*1*ν*2 + *<sup>s</sup>θν*1 + (1 − *<sup>θ</sup>*)*<sup>ν</sup>*2 (*s* + *<sup>ν</sup>*1)(*<sup>s</sup>* + *<sup>ν</sup>*2) . As *N* = 2, it is evident that there exists only one positive and real root *ρ*1 to the generalised Lundberg equation of Equation (6). Therefore, given also that *β*(*s*) = *θν*1 + (1 − *<sup>θ</sup>*)*<sup>ν</sup>*2, the ladder height distribution takes the form

$$\overline{H}(u) = \frac{1}{\Phi c^2} \left( \frac{\nu\_1 \nu\_2 - c\rho\_1(\theta \nu\_1 + (1-\theta)\nu\_2)}{-\rho\_1} \mathcal{T}\_{\overline{\rho}\_1} \overline{\mathcal{F}}\_X(u) + \frac{\nu\_1 \nu\_2}{\rho\_1} \mathcal{T}\_0 \overline{\mathcal{F}}\_X(u) \right),$$

which is in accordance with Li and Garrido (2005).
