**Proof:**

From theorem 1.5, we know that an *NHP* is an *MPP*. According to McFadden in ref. [9], if f g *N t*ð Þ, *t*≥0 is a *CP* with independent increments, then its transition intensities satisfy that *λ*0ð Þ¼ *t*; *a λ*1ð Þ *t*; *a* , but by expression (48), we get

$$\lambda\_0(t; a) = \frac{q}{\left(1 + \kappa t\right)^a} \neq \frac{a\kappa}{1 + \kappa t} + \frac{q}{\left(1 + \kappa t\right)^a} = \lambda\_1(t; a) \quad \text{if} \quad a \neq 0 \tag{61}$$

And therefore, *N t*ð Þ is a *CP* that does not have independent increments.

This was to be expected since that *MPP* has stationary increments but does not meet the condition of independent increments (see [23]).
