**4.2** *NHP as MPP*

The list of equivalences provided by Lundberg in ref. [18] is satisfied by the *NHP* defined in (6), which is presented in the following theorem:

Theorem 1.5: Let *N t*ð Þ be an *NHP* with marginal *pmf,* given by (5) and transition intensities, given by (21). Then:

$$\text{i. } \lambda\_n(t; a) \text{ satisfy } \lambda\_{n+1}(t; a) = \lambda\_n(t; a) - \frac{\lambda\_n'(t; a)}{\lambda\_n(t; a)} \text{ for } n = 0, 1, \dots$$

ii. *Pn*ð Þ*t* and *λn*ð Þ *t*; *a* satisfy the relation

$$\frac{P\_n(t)}{P\_{n-1}(t)} = \frac{t}{n} \lambda\_{n-1}(t; a) \quad \text{for} \quad n = 1, 2, \dots \tag{24}$$

## **Proof:**

i. By finding the derivative of function (21) with respect to *t*, we obtain

$$\begin{split} \lambda'\_n(t;a) &= -\left[ \frac{P\_0^{(n+2)}(t)P\_0^{(n)}(t) - P\_0^{(n+1)}(t)P\_0^{(n+1)}(t)}{\left(P\_0^{(n)}(t)\right)^2} \right] \\ &= -\frac{P\_0^{(n+2)}(t)}{P\_0^{(n+1)}(t)} \frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)} + \left(-\frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)}\right)^2 \\ &= -\lambda\_{n+1}(t;a)\lambda\_n(t;a) + \left[\lambda\_n(t;a)\right]^2 \end{split}$$

By dividing by *λn*ð Þ *t*; *a* , we have

$$\frac{\lambda'\_n(t;a)}{\lambda\_n(t;a)} = \lambda\_n(t;a) - \lambda\_{n+1}(t;a) \tag{25}$$

ii. By substituting (21) into (13), we get:

$$\frac{P\_n(t)}{P\_{n-1}(t)} = \frac{\frac{(-1)^n}{n!} t^n P\_0^{(n)}(t)}{\frac{(-1)^{n-1}}{(n-1)!} t^{n-1} P\_0^{(n-1)}(t)} = -\frac{t}{n} \frac{P\_0^{(n)}(t)}{P\_0^{(n-1)}(t)} = \frac{t}{n} \lambda\_{n-1}(t; a),$$

which completes the proof. □

In ref. [7], it is proved that the above three statements are equivalent. Corollary 1.5.1: Let *N t*ð Þ be an *NHP* with transition intensities given by (21), then

$$\frac{P\_n(t)}{P\_0(t)} = \prod\_{j=1}^n \frac{t\lambda\_{j-1}(t;a)}{j} \tag{26}$$

**Proof:**

Note that

$$\frac{P\_n(t)}{P\_0(t)} = \prod\_{j=1}^n \frac{P\_j(t)}{P\_{j-1}(t)}.$$

Substituting (24) in the above expression completes the proof. Corollary 1.5.2: Let *N t*ð Þ be an *NHP* with transition intensities given by (21), then

$$\prod\_{j=0}^{n-1} \lambda\_j(t; a) = (-1)^n \frac{P\_0^{(n)}(t)}{P\_0(t)} \qquad \qquad n \ge 1. \tag{27}$$

## **Proof:**

From (21), we get

$$\prod\_{j=0}^{n-1} \lambda\_j(t; a) = \prod\_{j=0}^{n-1} \left( -\frac{P\_0^{(j+1)}(t)}{P\_0^{(j)}(t)} \right) = (-1)^n \frac{P\_0^{(n)}(t)}{P\_0(t)}.$$

This finishes the proof of Corollary.

The following additional properties set in ref. [9] are also satisfied by *NHP*: Proposition 1.6: Let f g *N t*ð Þ; *t*≥0 be an *NHP* and Λ the continuous structure variable of the MPP. Then:

1. The transition intensities are such that

$$\mathbb{E}[\Lambda|\mathbf{N}(t)=n] = \lambda\_n(t; a). \tag{28}$$

and

$$\text{Var}[\Lambda|N(t) = n] = -\lambda'\_n(t; a). \tag{29}$$

2. The mean of *N t*ð Þ is given by

$$\mathbb{E}[\mathbf{N}(t)] = t\mathbb{E}[\Lambda]. \tag{30}$$

*Some Results on the Non-Homogeneous Hofmann Process DOI: http://dx.doi.org/10.5772/intechopen.106422*

3. The mean of Λ is given by

$$\mathbb{E}[\Lambda] = -P\_0'(\mathbf{0}).\tag{31}$$
