Assume $X_i$ are iid with mean 0 and variance $\sigma^2$ and $E(X^3_i) =0$.
Define $\bar{X}$ and $S^2 = \frac{\sum(X_i^2)}{n}-\bar{X}^2$.
Prove that convergence in Distribution of
$$ \sqrt n \left(\bar{X},S^2-\sigma^2\right)^T \rightarrow N\left(\boldsymbol{0},\Sigma\right) $$
where $\boldsymbol{0} = (0; 0)^T$ and $\Sigma$ has to be determined.
I am trying to use Cramer-Wald Theorem by evaluate characteristic function for linear combination of $\bar{X}$ and $S^2.$ So I start as the following
Let $\boldsymbol{a} = (a_1,a_2)^T$ and $T_n = \sqrt n \left(\bar{X},S^2-\sigma^2\right)^T$ and $Z_n = a_1\sqrt n\bar{X}+a_2\sqrt n\left(S^2-\sigma^2\right)$ then
$$ \phi_{Z_n}(t) = E\left\{\exp\left(it\sqrt n(a_1\bar{X}+a_2(S^2-\sigma^2))\right)\right\} =E\left\{e^{it\sqrt(n)a_1\bar{X}}e^{it\sqrt na_2(S^2-\sigma^2)}\right\} $$
and I got stuck here can you help me please