On deriving the asymptotic distribution of an estimator, I got stuck when proving something like follows:
Let $\{X_i(\beta):i=1,2,\ldots,n\}$ be i.i.d. with $\text{E}[X_i(\beta)]=\mu(\beta)$ and $\text{Var}[X_i(\beta)]=\sigma^2(\beta)$, then by Lindeberg-Levy we have $$ \sqrt{n}[X_i(\beta)-\mu(\beta)]\overset{A}{\sim}\mathcal{N}[0,\sigma^2(\beta)]. $$ However, we don't know what $\beta$ is, and thus we must find some estimator for that. Now, assume that we have some estimator for $\beta$ that follows $\hat\beta\overset{p}{\to}\beta$. Then it sure follows that $\text{E}[X_i(\hat\beta)]=\mu(\hat\beta)\overset{p}{\to}\mu(\beta)$ and $\text{Var}[X_i(\hat\beta)]=\sigma^2(\hat\beta)\overset{p}{\to}\sigma(\beta)$. My question is whether it still follows $$ \sqrt{n}[X_i(\hat\beta)-\mu(\beta)]\overset{A}{\sim}\mathcal{N}[0,\sigma^2(\beta)] $$ or not? In other words, is this $X_i(\hat\beta)$ still an efficient estimator for $\mu(\beta)$? Can I use Lindeberg-Levy or Khinchine's WLLN to prove it? I've tried for some hours but with no idea. It's just an intuition.
Any suggestion is appreciated, thanks!
Also, if you'd like to look into the original problem directly, please go to this question.