Let $d$ and $N$ be two large comparable integers, for example assume $$ N,d \to \infty, \quad d/N \to \gamma \in (0,\infty). $$ Let $w_1,\ldots,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \to \mathbb R$ be such that $|f(x)| \le \exp(cx^2/2)$, for all $x \in \mathbb R$ and for some $c < 1$. Define an $N \times N$ random matrix $T$ by $$ T_{ij} := \zeta_0(f_i)\zeta_0(f_j), $$ where the function $f_i:\mathbb R \to \mathbb R$ is defined by $f_i(x) := f(\|w_i\|x)$, and $\zeta_k(h) := \mathbb E_{G \sim N(0,1)}[h(G)]$ is the (probabilist's) $k$th Hermite coefficient of a function $h \in L^2(\mathbb R,N(0,1))$, with in particular $\zeta_0(h) := \mathbb E_G[h(G)]$.

I'm interested in a simple approximation of $T$ in terms of simple expressions in the $w_i$'s (e.g polynomial expressions). In this direction, one can obtain the following

(Easy case) If $\zeta_0(f)=0$, then $$ \|T-\mu\mu^\top\|_{op} = o_{d,\mathbb P}(1), \tag{1} $$ where $\mu \in \mathbb R^N$ is defined by $\mu_i := \zeta_2(f)(\|w_i\|^2-1)/2$.

The idea is to write $T_{ij} = D\mu\mu^\top D$, where $D$ is the diagonal matrix with $D_{ii} := \zeta_0(f_i)/(\zeta_2(f)(\|w_i\|^2-1)/2)$. Further, using the assumption that $\zeta_0(f) := \mathbb E_G[f(G)] = 0$, one writes

$$ D_{ii} := \mathbb E_G\left[\frac{\zeta_0(f_i)-\zeta(f)}{\|w_i\|-1}\right]\cdot\frac{1}{\zeta_2(f)(\|w_i\|+1)/2} $$

One can show that $$ \lim_{t \to 1}\dfrac{f(tG)-f(G)}{t-1} = \zeta_2(f) := \mathbb E_G[(G^2-1)f(G)]. $$ Combining with the fact that $\sup_{1 \le i \le N}||w_i|-1| = o_{d,\mathbb P}(1)$, we obtain the result (1).

Question. Without the condition $\zeta_0(f)=0$, is there a simple approximation of $T_0$ of the kind (1) ?