Let $X$ be an $n \times n$ random matrix with i.i.d. standard Gaussian entries, i.e., the entries $X_{ij}$ are independent standard normal random variables. Given any deterministic unit vectors $u, v \in \mathbb R^n$, calculate the mean and variance of $Z_n := u^\top X^\top Xu + \sqrt n v^\top Xv$, and prove the central limit theorem for $Z_n$ as $n\rightarrow \infty$.
This problem has a hint, saying that you can try moment method or Stein’s method, but I have no idea of how to use this. For the case $u^\top X^\top Xu$ you can just assume that $u=(1,0,\cdots,0)$ since multiplying a orthogonal matrix doesn’t changes the distribution of $X$, but I’ve no idea how can we proceed when we have a another vector $v$.
Can you provide any solutions/suggestions for me in this problem? Thank you!
How about, $$Xu \sim \mathcal N\left(0, \mathbf I_n\right)\quad \implies\quad u^\intercal X^\intercal Xu \sim \chi^2(n)$$
$$v^\intercal X v \sim \mathcal N(0, 1)$$
and $$\mathbb E\left[u^\intercal X^\intercal X u v^\intercal Xv\right] = 0 \quad \implies \quad \mathbb C\mathrm{ov}\left[u^\intercal X^\intercal X u, v^\intercal Xv\right] = 0$$ then $$\mathbb E\left[Z_n\right] = n\quad \text{and} \quad \mathbb V\left[Z_n\right] = 2n + n = 3n$$
and $$\frac{Z_n - n}{\sqrt{3n}} = \frac{u^\intercal X^\intercal X u - n}{\sqrt{3n}} + \frac1{\sqrt 3} v^\intercal Xv = \frac{\sqrt{2}}{\sqrt{3}}\frac{u^\intercal X^\intercal X u - n}{\sqrt{2n}} + \frac1{\sqrt 3} v^\intercal Xv \to \mathcal N(0, 1)$$