Given $X \in \mathbb{R}^{n \times p}$ where $X$ has linearly independent columns, suppose $$\frac{1}{n}X^TX \rightarrow A$$ where $A$ is a positive definite matrix.
I have two questions about the above statement:
(1) What does it mean for $\frac{1}{n}X^TX$ to converge to another matrix? Does it mean that I'll multiply each element in $X^TX$ by $1/n$, and each element will converge to a corresponding element in $A$?
(2) Why is it reasonable to assume that $A$ is a positive definite matrix? Does this have to do with the assumption that $X$ has linearly independent columns? If $X$ is not linearly independent, is it still possible for $\frac{1}{n} X^TX \rightarrow A$ where $A$ is positive definite?