I'm trying to solve this problem:
Let $\{X_i\}_{n \geq 1}$ i.i.d random variables with finite 4th-moment. Define:
$Y_n = {1 \over n}(X_1 + \cdots + X_n)$
$W_n = {1 \over n}(X_1^2 + \cdots + X_n^2)$
(a) Find the limit distribution of:
$\sqrt{n} \left(\left( \begin{matrix} Y_n \\ W_n \\ \end{matrix}\right) - \boldsymbol{\mu} \right)$
for an appropriate vector $\boldsymbol{\mu}$.
(b) Use the Delta Method for finding the limit distribution of:
$\sqrt{n} \left(\left( \begin{matrix} Y_n \\ W_n - Y_n^2 \end{matrix}\right) - \boldsymbol{\lambda} \right)$
for an appropriate vector $\boldsymbol{\lambda}$.
(c) Use the previous results to find the limit distribution of the coefficient of variation given by: $\sqrt{W_n - Y_n^2} \over Y_n$
I've been assuming that $E(X_1) = 0$, $E(X_1^2)=Var(X_1)=\sigma^2$.
For part (a) I found that for $\boldsymbol{\mu} = \left( \begin{matrix} 0 \\ \sigma^2 \\ \end{matrix}\right)$ you just need to find the matrix $\Sigma = \left( \begin{matrix} Var(Y_n) \quad Cov(Y_n, W_n) \\ Cov(Y_n, W_n) \quad Var(W_n) \\ \end{matrix}\right)$ and then use the central limit theorem. But I haven't been able to prove that $\Sigma$ is actually a constant matrix because the covariance part seems to depend on $n$.
Part (b) seems to be direct if you use the function $g(x,y) = (x, y - x^2)$. But again you need the matrix $\Sigma$.
And I havent't figured out how to use these results for part (c).
Any help would be really appreciated!