If $X_1, ... , X_n$ be iid exponential with mean $1/\lambda$. Let $S_n = X_1 + ... + X_n$.
a) Show that $S_n$ is $\Gamma(n, 1/\lambda)$.
Each $X_i$ is $\Gamma(1, 1/\lambda)$ by the definition of an exponential distribution. We can look at the mgf of $S_n$.
$$M(t) = E(e^{t \sum X_i}) = E(e^{tX_1})...E(e^{tX_n}) = (\frac{1}{1-t/\lambda})...(\frac{1}{1-t/\lambda}) = (\frac{1}{1-t/\lambda})^n,$$
which is the mgf of $\Gamma(n, 1/\lambda)$. So $S_n \sim \Gamma(n, 1/\lambda)$.
b) Show that $\frac{(S_n - n/\lambda)}{\sqrt{n}}$ converges in distribution and identify the asymptotic distribution.
By the Central Limit Theorem, we know that $\frac{(S_n - n\mu)}{\sqrt{n}(1/\lambda^2)} = \frac{(S_n/n - \mu)}{(1/\lambda^2)(1/\sqrt{n})}$ converges in distribution to $N(0,1)$.
We see that $\frac{(S_n/n - 1/\lambda)}{(1/\lambda^2)(1/\sqrt{n})} = \frac{(S_n/\sqrt{n} - \sqrt{n}/\lambda)}{(1/\lambda^2)}(\frac{\sqrt{n}}{\sqrt{n}}) = \frac{S_n - n/\lambda}{(1/\lambda^2)(\sqrt{n})}$
So, $$\frac{S_n - n/\lambda}{(1/\lambda^2)(\sqrt{n})} \xrightarrow{D} N(0,1)$$ implies that $$ \frac{S_n - n/\lambda}{(\sqrt{n})} = (1/\lambda^2)\frac{S_n - n/\lambda}{(1/\lambda^2)(\sqrt{n})} \xrightarrow{D} N(0, 1/\lambda^4).$$
c) Show that $\sqrt{n}(\sqrt{S_n/n} - \sqrt{1/\lambda})$ converges in distribution and identify the asymptotic distribution.
There is a theorem in our textbook (Introduction to Mathematical Statistics by Hogg) which states,
Let $\{X_n\}$ be a sequence of random variables such that $\sqrt{n}(X_n - \theta) \xrightarrow{D} N(0, \sigma^2)$. Suppose the function $g(x)$ is differentiable at $\theta$ and $g'(\theta) \not= 0$. Then $\sqrt{n}(g(X_n) - g(\theta)) \xrightarrow{D} N(0, \sigma^2(g'(\theta)^2)$.
From part b), we see that $$\frac{\sqrt{n}(S_n/n - 1/\lambda)}{(1/\lambda^2)} \rightarrow N(0,1).$$
So $$\sqrt{n}(S_n/n - 1/\lambda) \rightarrow N(0, 1/\lambda^4)$$.
Now we apply the theorem. We have $g(x)=\sqrt{x}$. So $g'(x)=1/(2\sqrt{x})$, and $g'(\frac{1}{\lambda}) = \frac{\sqrt{\lambda}}{2}$.
So by the theorem,
$$\sqrt{n}(\sqrt{S_n/n} - \sqrt{1/\lambda}) \rightarrow N(0, (\frac{\sqrt{\lambda}}{2\lambda^4})).$$
Do you think my answers are correct?