I am trying to understand a little of this thesis by Anna Rudas. In particular the continuous model presented in Section 2.2.2.
We are given a weight function $w: \mathbb{N} \rightarrow \mathbb{R}_{>0}$ and a Markovian pure birth process $X(t)$. The birth process has the property that $X(0) = 0$ and the birth rates $$ P(X(t + dt) = k+1 : X(t) = k) = w(k)dt + o(dt) $$
Rudas then defines the density $\rho$ of the point process corresponding to $X(t)$ to be $$ \rho(t) = \lim_{\epsilon \rightarrow 0} \frac{P((t,t+\epsilon) \text{ contains a point from $X$ } )}{\epsilon}. $$
Finally the formal Laplace transform of $\rho$ is given by $$\hat{\rho}(\lambda) = \int_0^\infty e^{- \lambda t} \rho(t) dt = \sum_{n=1}^\infty \prod_{i=0}^{n-1} \frac{w(i)}{\lambda + w(i)}.$$
**Question:**How is the far righthand side of $\hat{\rho}(\lambda)$ calculated?
Rudas states that the calculation is easy given that the intervals between successive jumps of $X(t)$ are ind. exponentially distributed variables with parameters $w(0),w(1),w(2),\dots$ respectively.