Suppose we have sample $y_1,y_2,...y_n$ from a borel distribution
$$P(Y=y;\alpha)= \dfrac{1}{y!}(\alpha y)^{y-1}e^{-\alpha y} , y=1,2..$$
The MLE of $\alpha$ is $\hat{\alpha} = 1-\dfrac{1}{\bar{y}} $
By using that $\Big(\bar{y} - \dfrac{1}{1-\alpha} \Big) \rightarrow N \Big(0,\dfrac{\alpha}{(1-\alpha)^3} \Big)$
How can I find the asymptotic distribution for $\hat{\alpha}$ (MLE)
This is a Property of MLE. \begin{equation} \sqrt{n}(\hat{\alpha}-\alpha)\xrightarrow{d}N(0,I^{-1}) \end{equation}
So,you just need to calculate the inverse of fisher information matrix $I^{-1}$.
https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Consistency More details could refer this link.