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
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