**Proof:**

See details in [2] or [10]. Note that from (14) we have that if *q* 6¼ 0 then:

$$\lim\_{t \to \infty} \frac{\mathbb{E}[\mathbf{N}(t)]}{t} = q. \tag{15}$$

It is possible from (14) to calculate the measure based on the variance-to-mean ratio (VMR) introduced by [11]:

$$ID(t) = \frac{Var[N(t)]}{\mathbb{E}[N(t)]} = 1 + a\kappa t.\tag{16}$$

As *ID t*ð Þ>1, then using the criterion of the VMR, we have that the *NHP* is an overdispersed *CP* and hence is an option for modelling over-dispersion in count data.

Using the expression (11), in **Table 1**, we present the functions for *qt* and *κt* that allow to obtain some *CP*. We consider the *CCPs* studied in [10], which are special cases of *NHP* when *a* ¼ 1 since this reduces to the Panjer counting process (see [12]). In addition, we consider other processes, such as the Neyman Type A process introduced by [13], the Poisson Pascal process introduced by [14] and the Pólya-Aeppli process introduced by [15].

### **3.1** *NHP* **is infinitely divisible**

The following relationships are identical to those of [16] which characterize infinitely divisible *pmf*:


#### **Table 1.**

*Functions qt and κt for some CCPs.*

Theorem 1.3: The *pmf P*f g *<sup>n</sup>*ð Þ*t* with *P*0ð Þ*t* >0 is infinitely divisible if and only if satisfies that

$$r(n+1)P\_{n+1}(t) = \sum\_{i=0}^{n} r\_i(t)P\_{n-i}(t) \qquad\qquad\text{for } t \text{ fixed.}$$

where the quantities *rn*ð Þ*t* with *n*∈ ℤ<sup>þ</sup> are nonnegative.

**Proof:** See details in [16].

Corollary 1.3.1: The *pmf P*f g *<sup>n</sup>*ð Þ*t* of the *NHP* is infinitely divisible. **Proof:**

By multiplying (12) by ð Þ *n* þ 1 we get

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

We denote

$$r\_i(t;a) = qt \binom{a+i-1}{i} \frac{\left(\kappa t\right)^i}{\left(1+\kappa t\right)^{a+i}} \qquad \qquad i = 0, 1, \ldots, n. \tag{17}$$

Note that *ri*ð Þ *t*; *a* ≥0, which allows to conclude that *Pn*ð Þ*t* is infinitely divisible.

The following relationship is given by [17]: all log-convex distributions are infinitely divisible but not all log-concave distributions are infinitely divisible.

Theorem 1.4: Let *N t*ð Þ be an infinitely divisible ℤþ-valued random variable with pmf *Pn*ð Þ*t* . Then

$$\mathbb{E}[N(t)] = \sum\_{i=0}^{\infty} r\_i(t; a) \tag{18}$$

### **Proof:**

We know that the expectation of *N t*ð Þ it is given by

$$\begin{aligned} \mathbb{E}[N(t)] &= \sum\_{n=1}^{\infty} n P\_n(t) = \sum\_{m=0}^{\infty} (m+1) P\_{m+1}(t) \\ &= \sum\_{m=0}^{\infty} \sum\_{i=0}^{m} r\_i(t; a) P\_{m-i}(t) \end{aligned}$$

Now, by interchanging the order of summation, we get

$$\begin{aligned} \mathbb{E}[N(t)] &= \sum\_{i=0}^{\infty} \sum\_{m=i}^{\infty} r\_i(t;a)P\_{m-i}(t) = \sum\_{i=0}^{\infty} r\_i(t;a) \sum\_{m=i}^{\infty} P\_{m-i}(t),\\ &\underset{j=m-i}{\equiv} \sum\_{i=0}^{\infty} r\_i(t;a) \sum\_{j=0}^{\infty} P\_j(t) = \sum\_{i=0}^{\infty} r\_i(t;a). \end{aligned}$$

which completes the proof.
