Questions on the Wishart distribution

Question

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

1. 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?

2. 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.

Answer 1

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 $d\vec{x}$.

Answer 2

$X^T X \to \infty$ as $n\to\infty$. But $X^T X / n \to \mathbb E (X_1^TX_1)$ almost surely by the Strong Law of Large numbers providing that the rows of $X$ are iid $p$-row vectors $X_i$ with $\mathbb E |X_i| <\infty$.

Note that $X^T X = \sum_{i=1}^n X_i^T X_i$.