Uncorrelated Martingales

Question

I'm not sure how to prove that Mn+1 - Mn & Mn are uncorrelated ?

If Mn = (Xn)^2 - 2nXn + n(n − 1); Where Xn is a Random walk Xn+1 = Xn +Yn+1 Where Yn - N(1,1) I already know this is a Martingale, but not sure how to go about it - is there a rule or an intuition?

Answer 1

$E(M_{n+1}-M_n) M_n=E(E(M_{n+1}-M_n) M_n)|\mathcal F_n)=EM_n(E(M_{n+1}-M_n)|\mathcal F_n)=0$ by martinagle property. Also, $E(M_{n+1}-M_n) EM_n=(0)(EM_n)=0$ so the cavariance is $0$.