Let $X_1, X_2, ... i.i.d$ and $Y_1, Y_2, ... i.i.d$ be the random variables s.t $E(X)=E(Y)=0$ and $Var((X,Y)^T)$ (which is 2 by 2 matrix) is positive definite. (not semi)
Let $A_n$ a $n \times 2$ random matrix
$\begin{bmatrix} X_1 & Y_1\\ X_2 & Y_2\\ ... &...\\ X_n & Y_n \end{bmatrix}$ .
does $P(rank ( A_n)=2) $ goes to $1$ as $n$ goes to the infinity?
I think the probability increases as n increases, but couldn't prove or disprove whether is goes to 1.
I also tried to use the law of large number ot $(A_n^TA_n)/n$ as it converges to $Var((X,Y)^T)$ under good conditions, but I don't know the relationship of matrix convergence and the rank of matrix.