Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying

• $\mbox{trace}(\Sigma_d/d)= 1$.
• $\|\Sigma_d\|_{op},\|\Sigma_d^{-1}\|_{op} = \mathcal O(d)$.

Define $R := Q\circ \overline{Q}$, the Hadamard (i.e element-wise) product of $Q:=XX^\top/d$ and $\overline{Q} := X\Sigma_dX^\top/d$.

Question. Does there exist absolute constants $a,b \in \mathbb R$ such that $\|R-R_0\|_{op} \overset{p}{\to} 0$, where $R_0 := a Q + b I_n$ ?

Solution to isotropic case

If $\Sigma_d = I_d$., then $\overline{Q} = Q$, and so $R = Q \circ Q=f(Q):=(f(q_{ij})_{i,j \in [n]}$, where $h:\mathbb R \to \mathbb R$ is defined by $f(t) = t^2$. By, Theorem 2.3 of El Karoui (2010), we know that $$ \begin{split} f(Q) &= \left(f(0)+\dfrac{f''(0)\mbox{trace}(\Sigma_d^2))}{2d^2}\right)1_n1_n^\top + f'(0)Q+(f(1)-f(0)-f'(0))I_n\\ & = \frac{1}{d^2}1_n1_n^\top + I_n. \end{split} $$ Thus, $\|R-I_n\|_{op} = o_{n,\mathbb P}(1)$.