The Question:
Given independent random variables $X_1, \dots,X_n \sim N(\mu,\mu^2)$, let $Q=\sum_{i=1}^n(X_i-\overline X)^2$ where $\overline X = \frac 1n \sum_{i=1}^nX_i$.
Use the delta method to find the asymptotic distribution of $\frac 13\Bigl(\overline X+2\sqrt{\frac{Q}{n-1}}\Bigr)$.
My Thoughts:
I understand how to use the delta method to find the asymptotic distribution for any function of only $\overline X$, but this is clearly not the case for $Q$.
Also, I suppose you can't let $Y_i=(X_i-\overline X)^2$ and then apply delta method with $\overline Y$ because the $Y_i$ are not independent. So, is the following approach correct?
Let $g(u)=\sqrt u$ (so $g'(u)=\frac{1}{2\sqrt u}$).
We know that $R:=\frac{1}{\mu^2}Q \sim \chi_{n-1}^2$, so
\begin{align} \ & \Bbb E (R)=n-1 \; \; \; \text{ and } \; \; \; \operatorname{Var}(R)=2(n-1) \\ \ \implies & g(R)=\sqrt R \approx N \biggl(g(\Bbb E(R)) \; , \; g'(\Bbb E (R))^2 \operatorname{Var}(R) \biggr)=N \Bigl(\sqrt{n-1} \; , \; \frac 12 \Bigr) \\ \ \implies & \sqrt Q = \mu\sqrt R \approx N \Bigl(\mu\sqrt{n-1} \; , \; \frac {\mu^2}{2}\Bigr) \end{align}
Mainly, I am not sure if we need to divide by $n$ in the asymptotic variance.