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

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 $h$ ?*

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).

---

Solution with added restriction that $h$ is Lipschitz continuous
---
>*Below, I do some computations which seem to suggest the result is true under some additional smoothness constraints on $h$. I'm not 100% sure of my calculations. I'd be grateful if someone else would double-check. Thanks in advance!*

So, suppose $h$ satisfies
- *(1) $h$ is Lipschitz-continuous.*
- *(2) $h$ is square-integrable w.r.t $N(0,1)$.*

>**Claim 1.** *Under the above assumptions, the following hold in the limit $d \to \infty$,*
$$
\begin{split}
|\alpha_d-a| &= \mathcal O(1/d),\\
|\beta_d-b| &= \mathcal O(1/d).
\end{split}
$$

Under the additional assumptions

- *(3) $h$ is smooth with Lipschitz-continuous gradient $h'$*,
- *(4) $h'$ is square-integrable w.r.t $N(0,1)$*,

we have

>**Claim 2.** $|\gamma_d-c| = \mathcal O(1/d)$ in the limit $d \to \infty$.


*Proof of Claim 1.* For any $p \in [1,\infty)$, define (when it exists!) the $L_p$ norm of a (real-valued) random variable $R$ by
$$
\|R\|_p := (E[R^p])^{1/p}.
$$

**Note.** If $R_1$ and $R_2$ are random variables, $\|R_1-R_2\|_p$ will always be understood w.r.t to an underlying [coupling][1] $(R_1,R_2)$ of $R_1$ and $R_2$.

- Now, let $(G_1,G_2) \sim N(0,I_2)$, independent of $X$. Let $W_p(X_1,G_1)$ be the Wasserstein distance of order $p$, between $X_1$ and $G_1$. For $W_2$-optimal couplings $(X_1,G_1)$ and $(X_2,G_2)$, one computes
$$
\begin{split}
&|\alpha_d-a| = |E[h(X_1)h(X_2) - h(G_1)h(G_2)]|\\
&\quad= |E[h(X_1)(h(X_2)-h(G_2))] + E[h(G_2)(h(X_1)-h(G_1))]|\\
&\quad\le |E[h(X_1)(h(X_2)-h(G_2))]| + |E[h(G_2)(h(X_1)-h(G_1))]|\\
&\quad\le \|h(X_1)\|_2\|h(X_2)-h(G_2)\|_2 + \|h(G_2)\|_2\|h(X_1)-h(G_1)\|_2\\
&\quad\lesssim \|X_2-G_2\|_2 + \|X_1-G_1\|_2\\
&\quad\lesssim W_2(X_2,G_2) + W_2(X_1,G_1)\\
&\quad\lesssim W_2(X_1,G_1).
\end{split}
\tag{1}
$$
The first inequality is the triangle inequality, the second is Cauchy-Schwarz, and the third is thanks to the Lipschitz continuity of $h$.
Recall the following fact (established here https://mathoverflow.net/a/401444/78539)

>**Fact 1.** $W_p(X_1,G_1) = \mathcal O(1/d)$ for all $ p \ge 1$.

We deduce from (1) that $|\alpha_d-a| = \mathcal O(1/d)$ as claimed.

- Next, we prove that $|\gamma_d-c| = \mathcal O(1/d)$. Indeed,
$$
\begin{split}
|\gamma_d-c| &= |E[h(X_1)^2 - h(G_1)^2| = |E[(h(X_1)+h(G_1))(h(X_1)-h(G_1)]|\\
&\le \|h(X_1)+h(G_1)\|_2\|h(X_1)-h(G_1)\|_2\\
&\le (\|h(X_1)\|_2+\|h(G_1)\|_2)\|X_1-G_1\|_2\\
& \lesssim \|X_1-G_1\|_2 = W_2(X_1,G_1) = \mathcal O(1/d).
\end{split}
$$
This completes the proof of **Claim 1.** $\quad\quad\Box$

*Proof of Claim 2.*
Consider the function $r:\mathbb R \to \mathbb R$ defined by $r(x) := xh(x)$. Let $(X_2,G_2)$ be an $W_2$-optimal coupling of $X_2$ and $G_2$ and let $(X_1,G_1)$ be a $W_4$-optimal coupling for $X_1$ and $G_1$. One computes
$$
\begin{split}
|\beta_d-b| &= |E[r(X_1)h'(X_2) - r(G_1)h'(G_2)]|\\
&= |E[r(X_1)(h'(X_2)-h'(G_2)) + h'(G_2)(r(X_1)-r(G_1))]|\\
&\le \|r(X_1)\|_2\|h'(X_2)-h'(G_2)\|_2 + \|h'(G_2)\|_2\|r(X_1)-r(G_1)\|_2\\
&\lesssim \|X_2-G_2\|_2 + \|r(X_1)-r(G_1)\|_2\\
&= W_2(X_2,G_2) + \|r(X_1)-r(G_1)\|_2\\
&= \|r(X_1)-r(G_1)\|_2+\mathcal O(1/d),
\end{split}
$$
where the first inequality is thanks to the Lipschitzness of $h'$. It only remains to bound the term $\|r(X_1)-r(G_1)\|_2$. To this end, note that 
$$
\begin{split}
\|r(X_1)-r(G_1)\|_2 &= |X_1h(X_1)-G_1h(G_1)|\\
&= \|X_1(h(X_1)-h(G_1))+h(G_1)(X_1-G_1)\|_2\\
&\le \|X_1\|_4\|h(X_1)-h(G_1)\|_4+\|h(G_1)\|_4\|X_1-G_1\|_4\\
&\lesssim \|X_1-G_1\|_4 = W_4(X_1,G_1) = \mathcal O(1/d).
\end{split}
$$
This completes the proof of **Claim 2.** $\quad\quad\quad\Box$


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