Let $X_1,X_2,\dots \,$ i.i.d. random variables with values in $\mathbb R^2$ and let $S_n:=\sum_{i=1}^n X_i, n\in\mathbb N. $ $U_A $ denotes the uniform distribution on A. Calculate the weak limes of $\frac{S_n}{\sqrt n}$.
$X_1=(Y_1,Y_2),Y_1,Y_2 \,$ i.i.d $\sim U\{-1,0,1\}$.
$\mathbb P(X_1=(-1,0))=\mathbb P(X_1=(1,0))=\frac13$; $\mathbb P(X_1=(-1,1))=\mathbb P(X_1=(1,-1))=\frac16$.
My approach according to 1:
We have to calculate $\operatorname E(X_1)\operatorname Var(Y_1),\operatorname Var(Y_2)$ and $Cov(Y_1,Y_2)$. Then determine the covariance matrix and can conclude by the central limit theorem the weak limes.
Thus $\operatorname E(Y_1)=\operatorname E(Y_2)=0$ and $\operatorname Var(Y_1)=\operatorname V(Y_2)=\frac23$.
Furthermore we obtain $$\operatorname Cov(Y_1,Y_2)=\operatorname E(Y_1Y_2)-\operatorname E(Y_1)\operatorname E(Y_2)=\frac13 \bigr((-1 \cdot 0)+(-1\cdot -1)+(-1\cdot1)\bigl )+ \frac13 \bigr((0 \cdot -1)+(0\cdot 0)+(0\cdot1)\bigl )+\frac13 \bigr((1 \cdot -1)+(1\cdot 0)+(1\cdot1)\bigl )-0=0$$ We obtain the covariance matrix $\Sigma$ given as
$ \begin{pmatrix} \operatorname Var(Y_1) & \operatorname Cov(Y_1,Y_2) \\ \operatorname Cov(Y_2,Y_1) & \operatorname Var(Y_2) \\ \end{pmatrix} $ $ =\begin{pmatrix} \frac23 & 0\\ 0 & \frac23 \end{pmatrix} $
Conditions for using CLT are fullfilled: $\cal L(\frac{S_n}{\sqrt n}){\overset{\text{d}}{\to}}\cal N(0,\Sigma)$.
My approach according to 2:
I will operate like in $1.$ I denote the first component of $X_1$ with $X_{1,1}$ and the second with $X_{1,2}$ $$\operatorname E(X_{1,1})= \frac13 \cdot-1 + \frac13 \cdot1 +\frac16 \cdot -1 +\frac16 \cdot 1=0$$ and analogous $$\operatorname E(X_{1,2})=0$$ Therefore $\operatorname E(X_1)=(0,0)$ and $$\operatorname Var(X_{1,1})=\operatorname E(X_{1,1}^2)=\frac13 -1^2 + \frac13 1^2 +\frac16 -1^2 +\frac16 1^2=1$$ and analogous $$Var(X_{1,2})=\operatorname E(X_{1,2}^2)=\frac13$$ Thus we obtain $\operatorname Var(X_1)=(1,\frac13)$. Now we have to calculate $\operatorname Cov(X_{1,1},X_{1,2})$.
$$\operatorname Cov(X_{1,1},X_{1,2})= \operatorname E(X_{1,1}X_{1,2})-\operatorname E(X_{1,1})\operatorname E(X_{1,2})=\operatorname E(X_{1,1}X_{1,2})= \frac13 (-1\cdot 0)+\frac13 (1\cdot 0) +\frac16 \cdot(-1\cdot 1)+\frac16 (1\cdot -1)=\frac{-1}{3} $$
Thus the covariance matrix is given by $ \begin{pmatrix} 1 & \frac{-1}{3}\\ \frac{-1}{3} & \frac{1}{3} \end{pmatrix}= \Sigma $
Again all conditions for using the CLT are fullfilled: $\cal L(\frac{S_n}{\sqrt n}){\overset{\text{d}}{\to}}\cal N(0,\Sigma)$.
Is this reasoning correct? Especially I am not sure about the correct form of the calculation of the covariance in $1.$ and splitting up the vector in $2.$ like I did. Some comments are welcome!