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 $c < 1$. Note that this implies $f \in L^2(\mathbb R,N(0,1)$. 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 $\zeta_k(h) := \mathbb E_{G \sim N(0,1)}[h(G)\mathrm{He}_k(G)]$ 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
$$
D_{ii} := \frac{\zeta_0(f_i)}{\zeta_2(f)(\|w_i\|^2-1)/2}.
$$
Further, using the assumption that $\zeta_0(f) := \mathbb E_G[f(G)] = 0$, one writes

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

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 that $\|D-I_N\|_{op} = o_{d,\mathbb P}(1)$, from which (1) follows.

**N.B.:** Here, $o_{d,\mathbb P}(1)$ is notation for a quantity which converges to zero in probability.

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