I'm need help with asymptotic normality of MLE.
Example:
$X_1,..., X_n$ iid $X_i$ ~ $Poisson(\lambda) $
Likelihood:
$L(\lambda)=\prod_{i=1}^{n} \frac{e^{\lambda}\lambda^{x_i}}{x_i!}$
Log-Likelihood:
$l(\lambda) = -n\lambda + \sum_{i=1}^{n}x_iln(\lambda)-\sum_{i=1}^{n}ln(x_i!)$
Score Function:
$U(\lambda)= -n +\frac{ \sum_{i=1}^{n}x_i}{\lambda}$
Fisher Information:
$I_{e}^{}(\lambda) = \frac{n}{\lambda}$
and
$ \frac{1}{I_{e}^{}} = I_{e}^{-1}(\lambda) = \frac{\lambda}{n}$
MLE:
$\hat{\lambda} = \sum_{i=1}^{n} \frac{x_i}{n}$
So far I understand and I know do the calculations. But for asymptotic normality I don't know if this it correct.
Asymptotic Normality:
$\sqrt{n}(\frac{\hat{\lambda}-\lambda}{Var(\hat{\lambda})})$ ~ $N(0, 1)$
$\sqrt{n}(\hat{\lambda}-\lambda)$ ~ $N(0, Var(\hat{\lambda}))$
$(\hat{\lambda}-\lambda)$ ~ $N(0, I_{e}^{-1}(\lambda))$
$(\hat{\lambda}-\lambda)$ ~ $N(0, \frac{\lambda}{n})$
$\hat{\lambda}$ ~ $N(\lambda,\frac{\lambda}{n})$
That's enough to show the asymptotic normality of $\lambda$?