What is the condition for $Var(\sqrt{n}(\hat\theta-\theta))\to V$ as $n\to \infty$, where $V$ is the asymptotic variance of $\sqrt{n}(\hat\theta-\theta)$. That is $\sqrt{n}(\hat\theta-\theta))\to^D N(0,V)$.
I ask this since in general convergence in distribution does not imply convergence in moments.