Questions on the Wishart distribution If $X$ is an $n\times p$ matrix where each row is iid multivariate normal, then $X^TX$ has a Wishart distribution.

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*What is known about the limiting distribution of $X^TX$ for large $n$ when the rows of $X$ are from some general multivariate distribution not necessarily normal?


*Is anything known about the distribution of the term $\mathbf{x}_0^T (X^TX)^{-1}\mathbf{x}_0$, where $\mathbf{x}_0\in\mathbb{R}^p$?
I'm not an expert in random matrix theory, but I would guess the assumption of normality in the definition on Wishart distribution isn't crucial perhaps due to some kind of central limit theorem. So Question 1 is really asking if we don't make the normality assumption, does $X^TX$ converge to some limiting distribution for large $n$? And that being a Wishart?
For question 2, it would seem that the distribution term $\mathbf{x}_0^T (X^TX)^{-1}\mathbb{x}_0$ should be very important since it appears in the variance of the predicted value in a linear regression when the regressors are assume to be random. But I can't find much about its distribution. The Wikipedia page on the inverse Wishart distribution shows a related expression with this on the denominator is chi-squared.
If anyone can point me to what is known about these questions even if it is not exactly what is asked, that would be helpful.

This isn't a complete answer, but I will give you a starting point to try to tackle part 2.
First notice that $\mathbf{x}_0^T (X^TX)^{-1}\mathbf{x}_0$ is a change of variables $\nu:\mathbb{R}^{n \times p} \mapsto \mathbb{R}$ which you can even more simply think of as mapping $V : \mathbb{R}^k \mapsto \mathbb{R}$ via a pairing function on the index set of the matrix $(X^TX)^{-1}$.
This vector-to-scalar transformation will have a joint density $f_{Y, X}(y, \vec{x}) = F_{X}(\vec{x}) \delta (y - V(\vec{x}))$ where $\delta$ is the Dirac delta distribution. And following this post, integrate the joint density with respect to $d\vec{x}$.
