Suppose $\hat{\theta}_n$ is the MLE for some parameter $\theta$. Suppose also that the MLE is such that the Cramer regularity conditions are fulfilled, and $\hat{\theta}_n$ is asymptotically normal with mean $\theta$ and variance equal to inverse of the Fisher information matrix. This convergence to normality is convergence in distribution, which does not imply convergence of moments. And yet this MLE is considered asymptotically unbiased (which presumably means that $E[\hat{\theta}_n]\to\theta$ as $n\to\infty$). How does one make this transition from asymptotic normality (with mean equal to $\theta$) to asymptotic unbiasedness (in the sense that $E[\hat{\theta}_n]\to\theta$ as $n\to\infty$)?
Thanks.
Adding a moment condition such as $E((\hat\theta_n)^2)\leqslant C$, indeed $E(\hat\theta_n)\to\theta$ follows.