limit of the probability of a random matrix has full rank

Question

Let $X_i$ be a $p$-variable i.i.d random vector that follows the distribution of a random vector $X$. not let $Y_n=(X_1,X_2,...,X_n)^T$ be a $n \times p$ random matrix. ($n>p$)

Is it true that if $E(XX^T)$ is a positive-definite matrix, then $P(rk(Y_n)=p)$ (the probability that $Y_n$ has full rank) goes to 1 as $n$ goes to $\infty$?

Answer 1

$E(XX^T)$ is not $>0$ iff there are $\alpha\in\mathbb{R},B\in \mathbb{R}^n\setminus \{0\}$, both non-stochastic, s.t. $P(B'X=\alpha)=1$.

In particular, if $E(XX^T)>0$ and $B\not= 0$, then $P(B^TX=0)<1$.