I'm pretty desperate : it's a simple question, but I don't manage to write the solution properly...

We have (a Galton-Watson process) \(\displaystyle Z_0=a\) and \(\displaystyle Z_n=\sum_{i=1}^{Z_{n-1}} X_{i,n}\), where the sequence \(\displaystyle (X_{i,n})_{i,n}\) is an iid sequence.

How can I show that \(\displaystyle Z_n\) is a homogeneous Markov Chain ? I know all the basic methods :

- prove that P(Z(n+1)=z(n+1)|Z0=a,Z1=z1,...,Zn=zn)=P(Z(n+1)=z(n+1)|Zn=zn) <--- I've tried it, but it wasn't rigourous enough to me

- prove that the process can be expressed as a function of Zn and an random variable U(n+1) independent with Zn <--- the problem here is that either Zn is in the index of the independent rv, either I have an infinite number of random variables. So I don't really know how to extend it...

Thanks