Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define another function $f:[-1,1] \to \mathbb R$ by
$$
f(q)=E[h(X_1)h(qX_1 + (1-q^2)^{1/2} X_2)].
$$

Examples
---
- If $h(z) \equiv \mathrm{sign}(z)$, then one can show that $f(q) \equiv \arcsin(q)/pi$.
- If $h(z) \equiv 1[z \ge 0]$, then $f(q) \equiv \arccos(-q)/(2\pi)$ 
- If $h(z) \equiv z$, then one can easily show that $f(q) \equiv q/2$. More generally, if $h(z) \equiv z^k$ (for an integer $z \ge 0$), then
$$
f(q) \equiv k!2^k (-i\cos\alpha)^k P_k(-i\tan\alpha),
$$
where $\alpha = \arcsin(q)$, $i = \sqrt{-1}$ is the imaginary unit, and $P_k$ is the $k$th Legendre polynomial.
- ...

The problem
---

Now, suppose $f$ is smooth around $0$ and define the coefficients
$$
\alpha_d := f(0),\,\beta_d := f'(0),\,\gamma_d := f(1),
$$
where we've emphasized the fact that these coefficients depend on the dimension $d$.


>
>**Goal.** *I'm interested in the limit of these coefficients when $d \to \infty$.*

To this end, further suppose $h$ is smooth on $\mathbb R$ and define more coefficients
$$
\begin{split}
a &:= (E[h(G)])^2 =\zeta_0(h)^2,\text{ with }G \sim N(0,1),\\
b &= (E[h'(G)])^2=(E[Gh(G)])^2 = \zeta_1(h)^2, \\
c &:= E[h(G)^2]=\zeta_0(h^2),
\end{split}
$$
where $\zeta_k(h)$ is the $k$th Hermite coefficient of $h$.

>**Question.** *In the limit $d \to \infty$, is it true that $\alpha_d \to a$, $\beta_d \to b$, and $\gamma_d \to c$ for any "sufficiently integrable" function $h$ ?*

Notes
---

- Of course, it might well be that in order to answer the above question in the affirmative, one needs some additional assumptions on the function $h$. I'm fine with this, as long as the additional conditions are as general as possible (for example, should cover Lipschitz-continuous functions, polynomials, etc. because these special cases can be established via "direct computation").

- The motivation of this question is that, after all, the marginal distribution of each coordinate of $X$ convergence to $N(0,1)$ (in both Wasserstein distance, say).

----

Update
---

See partial solution below (here https://mathoverflow.net/a/403398/78539), under additional Lipschitz continuity assumoptions.

  [1]: https://en.wikipedia.org/wiki/Coupling_(probability)