Branching process with martingale

Let be i.i.d. positive interger-value random variables with and . Define and recursively define

a) Show that is a martingale with respect to the filtration

b) Find and deduce that M_n has uniformly bounded variance if and only if

c) For , find

My proof so far.

a) This one is easy,

. So that proves that M_n is a martingale.

b) I'm having problem trying to break down this thing...

But how would I proceed from here? Thanks.

Re: Branching process with martingale

Hello,

You know the conditional variance : Var[Y|X]=E[Y^2|X]-E[Y|X]^2

This will give you E[Y^2|X], and the conditional variance is the sum of the variances, because the variables are independent. Sorry I don't use the latex, nor did I use the correct names for the variables, but it takes too much time to write them down :D