I need to show that the following is a martingale:
I know that $(X_i)_{i=0,1,...}$ is a sequence of i.i.d. r.v.'s with $E(X_i) = 0$ and $Var(X_i) = 4$. I need to show that $(S^2_n-4n)_{n=1,2,..}$ is a martingale (I remember that $S_n =$ sum of $X_i$).
So here I know that I need to show that $E(S^2_n-4n|F_{n-1})=S^2_{n-1}-4(n-1)$, where $F_n$ is the filtration.
Using independence, I get $E(S^2_n)-E(4n)$. I have the feeling that this $S^2_n$ should be a hint to use the info about the mean and the variance (because of the squared), but I don't see how I could do that.
Note that $Var(S_n)=E(S_n^2)$ since $S_n$ has expectation zero. Also, be aware that $S_n$ is not independent of $F_{n-1}$ since the first n-1 terms in the sum are measurable. So what you want is actually: $$E[S_n^2-4n|F_{n-1}]=E[(S_{n-1}+X_n)^2|F_{n-1}]-4n=E[S_{n-1}^2+2*S_{n-1}X_n+X_n^2|F_{n-1}]-4n$$ Now you can use that $S_{n-1}$ is $F_{n-1}$ measurable and $X_n$ is independent OF $F_{n-1}$. Can you finish the calculation from here?