# 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.

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.

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 $$d\vec{x}$$.
$$X^T X \to \infty$$ 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 $$X^T X = \sum_{i=1}^n X_i^T X_i$$.