# Does the product of three Gaussian random matrices converge in distribution to a Gaussian?

1. Suppose we have vectors $u,v \in \mathbb{R}^r$ with and matrix $W \in \mathbb{R}^{r \times r}$ where all entries of $u,v,W$ are iid $N(0,1)$. Does the following hold? $$\frac{1}{r} u^TWv \overset{d}\rightarrow N(0,1) \hspace{5mm}\text{as } r \rightarrow \infty$$

2. Extension: Suppose we have matrices $U,V \in \mathbb{R}^{r \times n}$ and $W \in \mathbb{R}^{r \times r}$ where all entries of $U,V,W$ are iid $N(0,1)$. Does the following hold? $$\frac{1}{r} U^TWV \overset{d}\rightarrow N(0,I) \hspace{5mm}\text{as } r \rightarrow \infty$$

Through numerical experiments I am convinced that these claims are true, but I'm having difficulty proving even 1. Expressing the LHS as a sum of $r^2$ terms, dependencies are introduced by the elements of $u,v$. Hence I was trying to use a version of the Central Limit Theorem for dependent variables, but without much success. Note that each entry of the LHS in 2 is the LHS in 1. Hence the extension 2, would be a matter of checking whether the entries of the LHS converge jointly to a Gaussian.

I would be grateful if someone could give a proof, or give pointers to where this problem is tackled.