**3. Basic properties of the** *NHP*

Theorem 1.2: Let *N t*ð Þ be an NHP then

i. The pgf of the process is given by

$$G\_{N}(z;t) = \begin{cases} \left(1 + \kappa(1-z)t\right)^{-q/\kappa} & \text{if} \quad a = 1\\ \exp\left\{-\frac{q}{\kappa(1-a)}\left[\left(1 + \kappa(1-z)t\right)^{1-a} - 1\right]\right\} & \text{if} \quad a \neq 1 \end{cases} \tag{11}$$

<sup>1</sup> We say that a function *g t*ð Þ with *<sup>t</sup>* <sup>∈</sup> <sup>ℝ</sup><sup>þ</sup> is completely monotonic if it has derivatives *<sup>g</sup>*ð Þ *<sup>n</sup>* ð Þ*<sup>t</sup>* for all *<sup>n</sup>* <sup>∈</sup><sup>ℕ</sup> and its derivatives have alternating signs, i.e., if ð Þ �<sup>1</sup> *<sup>n</sup> <sup>g</sup>*ð Þ *<sup>n</sup>* ð Þ*<sup>t</sup>* <sup>≥</sup>0, *<sup>t</sup>* <sup>&</sup>gt;0*:*

Note that *GN*ð Þ¼ *z*; *t P*0ð Þ ð Þ 1 � *z t* with 0≤ *z*<1.

ii. The *pmf* of *N t*ð Þ, for *t* fixed, satisfies the following recursive formula:

$$P\_{n+1}(t) = \frac{t\lambda(t;a)}{n+1} \sum\_{i=0}^{n} \binom{a+i-1}{i} \left(\frac{\kappa t}{1+\kappa t}\right)^i P\_{n-i}(t) \tag{12}$$

where *P*0ðÞ¼ *t GN*ð Þ 0; *t* is given by (9) and

$$P\_0^{(n+1)}(t) = \lambda(t; a) \sum\_{j=0}^n \binom{n}{j} (-1)^{j+1} \frac{\Gamma(a+j)}{\Gamma(a)} \left(\frac{\kappa}{1+\kappa t}\right)^j P\_0^{(n-j)}(t)$$

iii. If *a* ¼ 1 the *Pn*ð Þ*t* satisfies the recurrence relation

$$\frac{P\_{n+1}(t)}{P\_n(t)} = \frac{-t}{n+1} \frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)} = \frac{q+\kappa n}{1+\kappa t} \frac{t}{n+1}.\tag{13}$$

iv. The process *N t*ð Þ has a mean and variance given by

$$\mathbb{E}[\mathbf{N}(t)] = qt \qquad \text{and} \qquad \text{Var}[\mathbf{N}(t)] = (\mathbf{1} + a\kappa t)\mathbb{E}[\mathbf{N}(t)] \tag{14}$$
