Suppose $(X_1,Y_1),(X_2,Y_2),..$ are independent, identically distributed random vectors such that \begin{eqnarray*} P(X_1 = 2~and~Y_1=0) &=& 1/3 \\ P(X_1 = -1~and~Y_1=\sqrt{3}) &=& 1/3 \\ P(X_1 = -1~and~Y_1=-\sqrt{3}) &=& 1/3 \\ \end{eqnarray*} Use Central limit theorem in $\mathbb{R}^2$ to evaluate $$ \lim_{n \rightarrow \infty} P((X_1+..+X_n)^2 + (Y_1+..+Y_n)^2 \leq n). $$
The theorem statement is given in the following link. https://en.wikipedia.org/wiki/Central_limit_theorem#Multidimensional_CLT
Firstly, I calculate the expected value and it is a zero vector in this case, i.e, $\mu=(0,0)^t$ and the covariance matrix is $2I_2$.
By the Central limit theorem in $\mathbb{R}^2$, $$ \sqrt{n} (\bar{X_n}) \Rightarrow N(0,2I_2). $$ Now, $$ P(|\sqrt{n} (\bar{X_n})|^2>1)=P((X_1+..+X_n)^2 + (Y_1+..+Y_n)^2 > n) $$
I understand that by applying Chebyshev's inequality I get the bound of above term.
I am little confused at this step and not really sure how to use here $$ \sqrt{n} (\bar{X_n}) \Rightarrow N(0,2I_2). $$
Can anyone please help me with this thought?