Asymptotic distribution of MLE It has been well understood that the asymptotic distribution of MLEs of model parameters are jointly normally distributed.
But in many cases, the true value of model parameter is bounded. For example, to estimate the failure rate rho, the MLE is total number of failures divided by total number of items.
So in this case, how the normality assumption of MLE of the failure probability can be valid, given that normal distribution can take any values including negative values?

Asymptotic Normallity gives us an approximate distribution for the MLE when $n < \infty$
Under suitable conditions, as $n \to \infty$, $Var(\hat{\rho}) \to 0$. This means that for sufficiently large $n$, the weight given to invalid values (like negative values) becomes negligible. This is an approximate result, but it is a highly practical approximation in many circumstances. If the approximation is not good enough for you; you will need to do further work to derive an exact distribution of the MLE of interest.
An alternative approach, if you want a strictly valid MLE distribution is to construct the MLEs on an unconstrained scale, for example, since $\rho > 0$ you may prefer to form an MLE for $\log \rho$
