Can we use uncorrelated to prove independent?

Question

We have learnt in the course that the independence of two variables implies they are uncorrelated. i.e. $$p(X,Y)=p(X)p(Y)\Rightarrow E[XY]=E[X]E[Y]$$ We do know that the reverse doesn't hold. But can we use independence of all moments or even all statistics to get the independence? Does the following result holds? $$E[X^iY^i]=E[X^i]E[Y^i],\forall i=1,\cdots,\infty \Rightarrow p(X,Y)=p(X)p(Y)$$ $$E[f(X)g(Y)]=E[f(X)]E[g(Y)],\forall f,g\in\mathcal{F} \Rightarrow p(X,Y)=p(X)p(Y)$$