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).
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).