Prove that $Z =\sum_{j=1}^{n}S_{j}$ is distributed as $\mathrm{NB}(\frac{n\alpha}{\ln(1-\delta)},1-\delta)$

Question

Prove that $Z=\sum_{j=1}^{n}S_{j}$ is distributed as $$\mathrm{NegativeBinomial}\left(\frac{n\alpha}{\ln(1-\delta)},1-\delta\right)$$

Being $S_{j}=\sum_{i=1}^{N}X_{i}$ where $N$ is distributed as $\mathrm{Poisson}(\alpha)$ and $f_{X_i}(x)=\dfrac{-\delta^\epsilon}{\epsilon \log(1-\delta)}$ with $\epsilon=1,2,\dots$

The thing here is that i don't have any clue how to get the pdf of $S$, im guessing maybe with the MGF I could get something, but I´m not sure at all, a full lenght explanation would be much appreciated.

Answer 1

Hint I think that wikipedia gives a proof of the fact that the (Poisson) random sum of logarithmic distributions is NB distributed.

Then use the fact that the (non-random) sum of NB distributed RV's is NB distributed.