Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the unit-sphere in $\mathbb R^d$. Consider the quartic form $$ F := \frac{1}{m}\sum_{j,\ell=1}^m (w_j^\top w_\ell)(w_j^\top C w_\ell). $$

Question.What are good probabilistic lower and upper-bounds for $F$ only in terms of $\rho$ and the eigenvalues of $C$ ?

For example, the solution for the case where $C$ is **diagonal** will already be very helpful.

## Isotropic example

Thanks to this post https://mathoverflow.net/a/334219/78539, we know that if $C = (1/d) I_d$, then $F = m^{-1}\|WW^\top\|_F^2 = m^{-1}\sum_{j}\lambda_j(W W^\top)^2\overset{a.s}{\to} \langle \lambda^2\rangle_{\text{MP}(1/\rho)}$ (if I haven't made some scaling errors), where $\text{MP}(\gamma)$ is the Marchenko-Pastur law with parameter $\gamma$.

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