Asymptotic normality of a normal sample standard deviation ($\sqrt{n}(S-\sigma )\stackrel{d}{\to }N(0,{\sigma }^2/2)$)

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 $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ and the central limit theorem for independent squared standard normal random variables implyThe delta method for $g(v)=\sqrt{v}$, with derivative $g^{\prime }({\sigma }^2)=1/(2\sigma )$, 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.