I Just read a Paper where we have $(X_1, ..., X_n)$ independent RV with zero mean.
And there were an Expression like $$ Var = V + v $$ where $V:= \sum_{1=j}^n \sum_{1=i}^nE(X_j^2)E(X_i^2)$
and $v:= \sum_{1=i}^n(E(X_i^4)-3E(X_i^2)^2)$
And it was said that it is easy to show that v=o(V), so that for $n \to \infty$, $v$ is negligible.
But I have no idea how to handle the term $E(X^4)$ in generel.