I'm having an hard time at understanding how to explicitly calculate the distribution law of a branching process.
Let's call $(X_{i,j})_{i,j\,\geq\,1}$ a double set of indipendent random variables, all with the same discrete law $\nu$ such that $\nu(\mathbb{N})=1$ and $\int xd\nu(x)=\mu<\infty$, then we can define the branching process $(Z_n)_{n\,\geq\,0}$ such that $Z_0=1$ and $$Z_{n+1}=\sum_{i\,\geq\,1}X_{i,n+1}1_{\{i\,\leq\,Z_n\}}\quad,\quad\forall n\,\geq\,0$$ where $1_{\{i\,\leq\,Z_n\}}$ is the indicator function of the set $\{i\,\leq\,Z_n\}$.
Then we can pose the natural filtration for $Z_n$ and calculate $E(Z_n)=\mu^n$, so that $Y_n:=\frac{Z_n}{\mu^n}$ is a non negative martingale, which converges a.s. .
I don't know if I could say more about the a.s. convergence of $Y_n$, but that's not my question, my question is: now suppose we know $\nu(k)=p(1-p)^{k-1}$ for $k=1,2,...$ and for some $p\in(0,1)$, how can I explicitly calculate the distribution law of $Z_n$? Does this help me determine the $L^p$ convergence of the martingale $Y_n$?
