Sampling distribution of GBM Maximum-Likelihood estimator Given the geometric Brownian diffusion
$$ X_t = \mu X_t \, dt + \sigma X_t \, d W_t$$
I learnt that its maximum likelihood estimators are the following as this web article suggests
$$\hat \mu = \frac{\delta X}{\delta t} + \frac 12 \hat{\sigma}^2$$
$$ \hat{\sigma}^2 = - \frac 1N  \frac{(\delta X)^2}{\delta t} +  \frac{1}{\delta t} \sum_{n=1}^N \Delta X^2_n$$
However, I start wondering what distributions the estimators $\hat \mu$ and $\hat \sigma$ have? I know that $W_t$ and consequnetly $X_t$ are (log-) normally distributed, but does the same hold for the estimators?

It is common knowledge in asymptotic statistics that the asymptotic distribution of MLE in an exponential family is the normal distribution with mean being the MLE, and variance being the inverse fisher information.
