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.