If X1, X2, Xn are iids, this means that:
Var(X1+X2+...+Xn) = Var(X1) + Var(X2) +...+Var(Xn), right??
but does : Var(X1^2 + X2^2 + ... + Xn^2) = Var(X1^2) + Var(X2^2)+...+Var(Xn^2)?
If X1, X2, Xn are iids, this means that:
Var(X1+X2+...+Xn) = Var(X1) + Var(X2) +...+Var(Xn), right??
but does : Var(X1^2 + X2^2 + ... + Xn^2) = Var(X1^2) + Var(X2^2)+...+Var(Xn^2)?
If are independent then so are . To see that, take and let . We have
and since are independent, we have and we are done.
We conclude that your formula is true if exists (but the problem doesn't come from the independence).