In a proof I have seen, a linear regression model of the following form was assumed:
$Y = X\beta + \epsilon$
Where $\epsilon \sim N(0, \sigma^2I)$. The proof involved looking at the distribution of the MLE estimate of $\sigma^2$ as $n$ went to infinity. The author had assumed that as $n$ grew large, the estimate of $\sigma^2$, $\hat{\sigma}^2$, followed a central limit theorem (i.e. Eventually became normally distributed with some variance).
This left me thinking - if you assume a central limit theorem for $\hat{\sigma}^2$, could you also assume a central limit theorem for $\hat{\sigma}$? i.e. Is there any reason that it would be more valid to assume a central limit theorem for the variance estimate than the standard deviation estimate or vice versa? Is there any theory that supports taking either of these and in general which is the more appropriate assumption for the linear regression model?