I understand that $$\left[I(\theta_0)\right]^{1/2}(\hat{\theta}-\theta_0)\overset{d}{\rightarrow}N(0,1),$$ where $I(\theta_0)$ is the Fisher information at $\theta_0$, $\hat{\theta}$ is a MLE of $\theta$.
My note says $I(\theta_0)$ can be replaced by $I(\hat{\theta})$ justified by Slutsky's theorem. Here is my proof: $$I(\hat{\theta})^{1/2}(\hat{\theta}-\theta_0)=I(\hat{\theta})^{1/2}I(\theta_0)^{-1/2}I(\theta_0)^{1/2}(\hat{\theta}-\theta_0)\overset{d}{\rightarrow}N(0,1),$$ since $I(\hat{\theta})^{1/2}I(\theta_0)^{-1/2}\overset{p}{\rightarrow}1$, we have $$\left[I(\hat{\theta})\right]^{1/2}(\hat{\theta}-\theta_0)\overset{d}{\rightarrow}N(0,1)$$
Is this correct? If not, what part is wrong?
In this proof, I used $I(\hat{\theta})^{1/2}I(\theta_0)^{-1/2}\overset{p}{\rightarrow}1$. This hold when $\hat{\theta}\overset{p}{\rightarrow}\theta_0\implies I(\hat{\theta})^{1/2}\overset{p}{\rightarrow}I(\theta_0)^{1/2}$. Is $I(\theta)$ always continuous for it to hold, or the continuity of $I(\theta)$ is an assumption we need to make?
Your proof is correct. Also, $I(\theta)$ is not always continuous, so you have to assume that it is continuous (only needed to be continuous close to $\theta$) in order to use Slutzky's Theorem.