**5.1 Other expressions for Pn**ð Þ**t in terms of** *λ***n**ð Þ **t; a**

Theorem 1.7: Let *N t*ð Þ be an *NHP* with transition intensities given by (21), then

$$P\_n(t) = Q\_n(t) - Q\_{n+1}(t) \quad \text{for} \quad n \ge 1,$$

where *Q*0ð Þ*t* is Heaviside's step function and

$$Q\_{n+1}(t) = \int\_0^t \lambda\_n(v; a) P\_n(v) dv. \tag{35}$$

**Proof:**

We write the expression (22) as

$$\frac{d[P\_n(\tau)]}{d\tau} = \lambda\_{n-1}(\tau; a)P\_{n-1}(\tau) - \lambda\_n(\tau; a)P\_n(\tau) \qquad \text{for} \qquad n \ge 1.$$

By integration of the above expression with respect to *τ* between 0 and *t*, we get

$$\begin{aligned} \int\_{0}^{t} d[P\_{n}(\tau)] &= \int\_{0}^{t} \lambda\_{n-1}(\tau; a) P\_{n-1}(\tau) d\tau - \int\_{0}^{t} \lambda\_{n}(\tau; a) P\_{n}(\tau) d\tau\\ &P\_{n}(\tau)|\_{0}^{t} = Q\_{n}(t) - Q\_{n+1}(t) \qquad \text{ for} \qquad n \ge 1. \end{aligned} \tag{36}$$

Since *Pn*ð Þ¼ 0 0, ∀*n*≥1, so the proof is completed.

Corollary 1.7.1: Let *N t*ð Þ be an *NHP* with transition intensities given by (21), then

$$P[N(t) > n] = Q\_{n+1}(t) \qquad \text{ for} \qquad n \ge 0 \tag{37}$$

**Proof:** The proof consists of a direct calculation

$$\begin{aligned} P[N(t) > n] &= \mathbf{1} - P[N(t) \le n] \\ &= \mathbf{1} - \sum\_{j=0}^{n} P\_j(t) = \mathbf{1} - P\_0(t) - \sum\_{j=1}^{n} P\_j(t) \end{aligned}$$

Using the previous result:

$$\begin{split}P[N(t) > n] &= \mathbf{1} - P\_0(t) - \sum\_{j=1}^{n} \left[ Q\_j(t) - Q\_{j+1}(t) \right] \\ &= \mathbf{1} - P\_0(t) - \left[ Q\_1(t) - Q\_{n+1}(t) \right] \end{split} \tag{38}$$

Note that

$$Q\_1(t) = \int\_0^t \lambda\_0(v; a) P\_0(v) dv = -\int\_0^t P\_0'(v) dv = -P\_0(v)|\_0^t = \mathbf{1} - P\_0(t)$$

Replacing *Q*1ð Þ*t* in (38) the proof is completed. The expression (37) allows to calculate the *cdf* of an *NHP*. Corollary 1.7.2: The function *Qn*þ<sup>1</sup>ð Þ*t* satisfies the following condition:

$$\lim\_{t \to \infty} Q\_{n+1}(t) = \mathbf{1} \qquad \text{for} \qquad n \ge 0. \tag{39}$$

**Proof:**

From (37), we get

$$\lim\_{t \to \infty} Q\_{n+1}(t) = \lim\_{t \to \infty} \left[ 1 - \sum\_{j=0}^{n} P\_j(t) \right].$$

As we have for *n* ≥1 : *Pn*ð Þ¼ ∞ 0, and using the above relationship

$$\lim\_{t \to \infty} Q\_{n+1}(t) = 1 - \lim\_{t \to \infty} P\_0(t).$$

For example, from expression (9) when *a* ¼ 1, we have:

$$P\_0(t) = \left(1 + \kappa t\right)^{-\frac{q}{\kappa}} \qquad \text{ for } \qquad \frac{q}{\kappa} > 0 \tag{40}$$

and we take the limit as *t* ! ∞, we get:

$$\lim\_{t \to \infty} Q\_{n+1}(t) = \mathbf{1} - \lim\_{t \to \infty} (\mathbf{1} + \kappa t)^{-\frac{q}{\kappa}} = \mathbf{1} \cdot \square$$

Proposition 1.8: Let *N t*ð Þ be an *NHP* with transition intensities given by (21), then

$$\exp\left\{-\int\_{t}^{t+h} \lambda\_n(v;a)dv\right\} = \frac{P\_0^{(n)}(t+h)}{P\_0^{(n)}(t)}\qquad\qquad\text{for}\quad h \ge 0.\tag{41}$$

**Proof:**

By substituting (28) into (40), we have

$$\begin{aligned} \left\{ \exp \left\{ - \int\_{t}^{t+h} \lambda\_n(v; a) dv \right\} = \exp \left\{ \int\_{t}^{t+h} \frac{P\_0^{(n+1)}(v)}{P\_0^{(n)}(v)} dv \right\} \\ &= \exp \left\{ \int\_{t}^{t+h} d \left[ \ln \left( P\_0^{(n)}(v) \right) \right] \right\} \\ &= \exp \left\{ \cdot \ln \left[ P\_0^{(n)}(v) \right] \Big|\_{t}^{t+h} \right\} = \frac{P\_0^{(n)}(t+h)}{P\_0^{(n)}(t)}. \end{aligned}$$

Corollary 1.8.1: Let *N t*ð Þ be an NHP. If the probability that no event occurs in a small interval of length *h* is denoted by *P*0ð Þ *t*, *t* þ *h* , that is *P*0ð Þ¼ *t*, *t* þ *h PNt* ð Þ ð Þ� þ *h N t*ðÞ¼ 0 , then

$$P\_0(t+h) = P\_0(t) \cdot P\_0(t, t+h) \qquad \qquad \text{for} \quad t, h \ge 0. \tag{42}$$
