Hi guys !
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) and , where the sequence is an iid sequence.
How can I show that 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...