For all $n \in \mathbb{R}$ there is $(x_{n,i})_{i \in \mathbb{n}}$ a sequence of independently and identically distributed variables with $P(X_{1,1} \in \mathbb{N}) = 1$ and $\mu := \mathbb{E}(X_{1,1}) < \infty$. let be $(Z_n)_{n \in \mathbb{N}}$ and adapted process with $Z_0 = 1$ and $Z_{n+1}=\sum_{i=1}^{z_n} X_{n,i}$
Determine $\mu \in \mathbb{R}$ so that $(Z_n)_{n \in \mathbb{N_0}}$ a martingal is and give the $Z_n$ is a Galton-Watson process and the Doob decompostion is $Z_n = M_n + A_n, n \in \mathbb{N_0}$ where $(M_n)_{n \in \mathbb{N_0}}$ is a martingal and $(A_n)_{n \in \mathbb{N_0}}$ is a predictable process with $A_0$. I'm totally struggling at this point and I would really appreciate your help a lot. $E[Z_{n+1}]=E[X_{1,n+1}+X_{2,n+1}+\ldots+X_{Z_{n},n+1}]= \sum_{k=0}^\infty P(Z_n=k)E[X_{1,n+1}+X_{2,n+1}+\ldots+X_{n+1}] =\sum_0^\infty P(Z_n=k)k\mu=\mu\sum_{k=0}^\infty P(Z_n=k)k=\mu E[Z_n]E[Z_{n+1}|F_n]=E[X_{1,n+1}+X_{2,n+1}+\ldots+X_{Z_{n},n+1}|F_n]$ does it make sense what I wrote it above?