= Asymptotic normality of a normal sample standard deviation
{title2=$\sqrt n(S-\sigma)\xrightarrow dN(0,\sigma^2/2)$}
For a <normal sample> of <variance> $\sigma^2>0$, define $S^2=(n-1)^{-1}\sum_i(X_i-\overline X)^2$. The exact <chi-squared distribution> identity $(n-1)S^2/\sigma^2\sim\chi^2_{n-1}$ and the <central limit theorem> for <independent> squared <standard normal random variables> imply
$$
\sqrt n(S^2-\sigma^2)\xrightarrow dN(0,2\sigma^4).
$$
The <delta method> for $g(v)=\sqrt v$, with <derivative> $g'(\sigma^2)=1/(2\sigma)$, therefore gives the displayed limit for $S$. Exact <independence of the normal sample mean and sample variance> also makes $S$ independent of $\overline X$. The resulting joint limit supports combining location and scale estimates when a model relates their parameters.
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